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U(E) is a representation, the induced representation re : G --> U(L2 (G/G0, E ) ) is given by (re(g)f)(x) = p(x, g) 1 12 cx(x, g)f(xg) where ex is a cocycle corresponding to 0' For finite groups, induced representations were studied by Frobenius, and the general theory for locally compact groups was developed by [Mackey 1]. We illustrate the ideas of this section by returning to an unfinished point in section 4 1 , namely the proof of 4.1 . 6(a).
4.2.20 Proposition: If G is amenable and H c G is a closed subgroup, then H is amenable.
Proof: Suppose 0' : H --> Iso(E) is a representation where E is a separable Banach
space and A c E * is an affine H-space . Form the adjoint of the induced repre sentation, re*, of G on L00(G/H, E *). Let A = {
H/1 n H 0 By Example 4..2J 8, a is equivalent (as a cocycle into H) to a cocycle f3 taking values in 1 n H0. Thus, to complete the proof of assertion (i) of Lemma 5 2. 7, it suffices to prove assertion (ii).. Choose a function cp:S -. H such that for each g, f3 (s, g) - 1 cp (s)a (s, g) = cp (sg) a. e. Thus, qi (s) = qi (sg) where qi is the composition of cp with the projection H --> 1\H/1. Since 1\H/1 is finite and G is ergodic on S, qi is constant on a Proof of Lemma 5.2.7: The cocycle
102
Ergodic theory and semisimple groups
conull set, say
l such that cp(s) =
is a-invariant We now claim that the action of H on Cfl is smooth. If A, B E Cfl and hn E H with A h n --> B, then since A, B, K are compact it follows that { hn} lies in a compact subset of H. Thus, we can assume hn --> h E H, and so B = A h. Thus, all H-orbits on Cfl are closed, verifying smoothness. Since X/H is essentially invariant X/H is countably separated by the smoothness assumption and since G is ergodic on S, ifJ is essentially constant Thus, there is a point x E X such that for almost all s, H/H x . The result then follows from Example 4..2.18(b). 1 ( Y, v) is a measure class preserving map, we obtain a map B(X ) which is injective, and B(X ) is an injedive continuous Boolean operation preserving map, then there is an a.e. defined measure class preserving map ( Y, v) such that Y such that B = Y is a measure class preserving G-map, then B(X ) is a continuous Boolean G-map, then there is a conull G-invariant subset X 0 c X and a measure class preserving G-map Y such that B(G/P) is now a G-map, there is a measure class preserving G-map t/1 : G/P ---> X, and since t/1 * = G/P', we can suppose (replacing P' by a conjugate subgroup if necessary) that P c P' . Finally, since the restriction of the G-action on X to r is the same a.e.. as the original r-action, (and in particular is isomorphic to the original r-action), we have X � G/P' as r-spaces, verifying 8.1 . 3. 0 Then for almost all x E A, T"(x) E A for infinitely many positive n. 00 in r, such that e - 1(x), e - 1(x)yn E 8 - !(qy; 1 (B)). Let lj; = qy o 8 Then the same calculation as in the beginning of the proof of Lemma 9. 3. 3 shows that 1 1 lj; (8 - 1(x) y n) = lj; (8 - (x) )Yn · However, e - 1(X) }'n E 8 - 1(qy; (B)), so lj;( 8 - 1(x) y n) E B for all n . O n the other hand, since H , c G is finite, if Y n E r, }'n ---+ 00 , then for any y E GjH,, YYn ---+ oo, i e , leaves every compact subset of G/H, In particular, this is true for y = lj; (8 - 1 (x) ), which is a contradiction . This proves Lemma 9.3 .5. =
Remark: The reader will observe throughout the proof of Theorems 5 . L2 and 5.2..5 the constant interplay between ergodicity of certain actions and
smoothness of other actions. One way in which this was used was to show that a cocycle was equivalent to one that took values in a subgroup. This type of result is generally useful, and we can formalize the technique we have employed in the following..
(Cocycle Reduction Lemma). Suppose a:S x G -> H is a cocycle into a locally compact group H where G acts ergodically on S. Suppose X is a continuous H-space on which H acts smoothly. lf there is an a-invariantfunction
Lemma:
Proof: We have for each g,
Rigidity
109
We now discuss some examples of ergodic actions of semisimple groups with finite invariant measure . We have already seen one source of examples in Example 22. 14. Namely, if G is a connected senisimple Lie group with no compact factors, G c H where H is also semisimple, then the action of G on Hjr will be an irreducible ergodic action if r c H is an irreducible lattice Another source of examples we have seen is actions induced from actions of lattices (see, e.g the proof of Corollary .5 . 2. 2).. We applied this in Corollary .5 . 2.3 to the action of SL (n, Z) on IR"/Z" Another important class of ergodic actions of lattices (and hence by inducing, actions of simple groups) is actions defined by embedding the lattice in a compact group.
Example: Let G be a semisimple algebraic Q-group, and r = Gz n G2, so that r is a lattice in G2 Then r has a number ofhomomorphisms into compact groups . For example if Zp is the compact commutative ring of p-adic integers, then for each p we have a homomorphism r � SL (n , Zp). Similarly, via the map Z � Z/pZ, we obtain homomorphisms r � SL (n , Z/pZ) for each p, and hence homomorphisms into products of groups of this type. Another standard arith metic construction yields homomorphisms into compact Lie groups . For example, fix positive integers m, n, n < m, and consider the quadratic form 5.2.1 2
m-n
L: x? - J2
i=l
m
I
i=m-n+ l
x? Let O (m, n) be the subgroup of GL(m, C) leaving this
form invariant Then O (m, n) is a simple algebraic group defined over the field Q (j2).. Let (J be a field automorphism of (: such that (J (j2) = - J2. This induces a map (J:GL(m, C) � GL(m, C), and CJ (O(m, n)) is then clearly the group leaving invariant the quadratic form
m-n
L: x? + J2
i=l
m
L:
i = m -n + l
x f . Thus, (J(O(m, n))IR
is a compact Lie group. Let r = O(m, n)z[v' 2l· Then (J:r � (O(m, n) )IR is an injec tive homomorphism into a compact Lie group . Furthermore, as we shall discuss further in Chapter 6, one can show that r is a lattice in the Lie group O(m, n)IR, and that IR-rank (O(m, n)) = min(n, m - n). By taking the closure of the image of r, we thus see that there are a number of examples of homomorphisms cp:f � K, K compact, and cp (r) dense in K. It follows from Lemma 2..21 3 that r acts ergodically on K, and hence on K/Ko where K 0 c K is any closed subgroup. We remark that if G is a connected semisimple Lie group without compact factors, and r c G is an irreducible lattice, one can show that the action of G obtained by inducing an ergodic action of r of the above type is actually an irreducible ergodic action of G [Zimmer 1 3] . That this is not true for arbitrary ergodic actions of r can be seen from the following example. Let r c G 1 x G 2 be an irreducible lattice, G; non-compact Let S be an ergodic G 1 space with finite invariant measure . The
Ergodic theory and semisimple groups
1 10
projection of r onto G 1 is dense by the irreducibility assumption, and hence r acts ergodically on S (by Corollary 2. 2.1 7, for example) By Proposition 4.2. 22, the induced G 1 x G 2 action is just the product G 1 x G 2 action on S x (G 1 x G 2 )jr However, G2 does not act ergodically on this space, so the G 1 x G 2 action is ergodic but not irreducible. Of course, for actions of lattices in simple groups, this question does not arise 5.2.13 Example: We shall now sketch another construction of finite measure preserving ergodic actions of semisimple Lie groups . For each such G to which the rigidity theorem (52. 1 ) applies this will provide uncountably many mutually non-orbit equivalent, essentially free, finite measure preserving ergodic actions. Let Jf be a real (separable) infinite dimensional Hilbert space, and let { e;} be an orthonormal basis. Let v be the Gaussian measure on IR, i.e , dv
(2n) - 1 e - x21 2 dx. Let 0
00
00
II IR and J1. the product measure J1. = Ilv. We 1 1 have a linear map T: Jf -+ L2 (0) given by T(e;) = p; where p; : O -+ IR is the projection onto the i-th factor.. Then T is an orthogonal isometry of Jf and T(Jf). =
=
By the theory of Gaussian random processes (see [Kuo 1 ] for example), for each orthogonal operator U on Jf, there exists an (essentially unique) a. e. defined measure preserving transformation fu : 0 -+ 0 such that the induced map i t U(O, IR) -+ L2 (0, IR) makes the following diagram commute: T
Jf ----� £ 2 (0)
Vl Jf ------:::;. £2 (0) . T
If G is locally compact and n is an orthogonal representation of G on Jf, then for each n(g) we obtain an a. e. defined transformation f"(g), and by Appendix B, we can choose these so as to define an action of G on 0 preserving the measure J1. Furthermore, letting Tc be the complexified map Jfc -+ L 2(0, C) we see that the complexification of the original representation n is a subrepresenta tion of a, the representation of G on L2 (0, C) defined by translation. I n fact, one can show that
a �
00
2:87 S"(nc)
n=O
where S"(nc) is the n-th symmetric power
of the complexification of n . From this, one readily deduces [Segal 1 ] that the action of G is ergodic if and only if n has no finite dimensional invariant subspaces. Suppose now that n1 , n 2 are infinite dimensional irreducible unitary repre sentations. Viewing them as orthogonal representations, we then obtain two
Rigidity
111
probability measure preserving ergodic actions of G. If these actions are equivalent, then (n 2 )c is a direct summand of S"((n i )c) for some n. Thus, for any given n 1 , there can be at most countably other irreducible representations which will define the same ergodic action via this construction. Thus, if G is any locally compact group with an uncountable number of inequivalent irreducible infinite dimensional unitary representations, G has uncountably many inequivalent measure preserving ergodic actions. Suppose now that n 1 , n 2 as above but that the actions defined by n1 and n 2 are equivalent modulo an automorphism A of G. (That is, if G acts on Q, define a new action by (s, g) ---+ s A(g).. Equivalence modulo A means equivalence after modifying one action in this fashion.) Then (n 2 )co A is a direct summand of S"( (n dc) for some n. If A is an inner automorphism, then n 2 A � n 2 Thus, if G has uncountably many inequivalent infinite dimensional unitary repre sentations, and Out(G), the outer automorphism group, is finite, then G has uncountably many probability measure preserving transformations no two of which are equivalent modulo an automorphism of G. Since noncompact semi simple Lie groups have this property, one then deduces from the rigidity theorem (5.2. 1) that if G is as in 52. 1 , G has uncountably many mutually non-orbit equivalent finite measure preserving ergodic actions. The rigidity theorem for ergodic actions (52J) and Corollary 5..2. 2 give us a good deal of information about orbit equivalence for actions of lattices with finite invariant measure. However, some of the most natural actions of lattices do not have finite invariant measure, for example the action of SL(n, Z) on IR", lP" - \ and other Grassmann or flag varieties. For the action on the variety of full flags, the action will be amenable and hence the results of Section 4.3 apply. (See Example 4.31 8).. Here we wish to indicate without proof that one can extend the techniques of this chapter to obtain information on orbit equivalence for many natural actions of lattices without invariant measure. We recall that any locally compact group H has a unique maximal normal amenable subgroup N (Proposition 4J J 2). If H is a connected Lie group, then by Corollary 41 . 9, H/N will be a connected semisimple Lie group with trivial center and no compact factors. Hence, if H is almost connected (i..e. , H/H 0 is finite) the connected component of the identity of H/N will be a semisimple group of this type as well. a
Theorem [Zimmer 9]: Let G;, i = 1, 2, be connected semisimple Lie groups with finite center and no compact factors, and let r; c G; be an irreducible lattice. Let H; c G; be an almost connected non-compact subgroup and N; c H; the maxi ma/ normal amenable subgroup.. Assume IR -rank(H dN 1 ) � 2. If f; acts essentially 5.2.1 4
Ergodic theory and semisimple groups
1 12
freely on G;/H; and the r 1 action on GdH 1 is orbit equivalent to the rz action on G 2 / H 2 , then H d N 1 and H z/ N 2 are locally isomorphic. H ::::l Z( G) (the center of G), and H does not contain a normal non-central subgroup of G, then r;r n Z( G) acts essentially freely on G/H To see this, observe that for y E r, {xE G/H i x y = x} is a closed subvariety, and hence if it is of positive measure, y fixes all of G/H, which implies yE Z(G). Essential freeness then follows from 5..2.4. Remark: If G, H are (real points of) algebraic groups,
Corollary [Zimmer 9]: (a) As we vary n, n � 2, the actions ofSL(n, Z) on lP" - 1 (1R) are mutually non-orbit equivalent. (b) As we vary n, n � 2, the actions of SL(n, Z) on IR" are mutually non-orbit equivalent. (c) For a fixed n � 4, let Gn k be the Grassman variety of k-planes in IR". Then as k varies, 1 � k � [n/2], the actions of SL(n, Z) on Gn k are mutually non-orbit equivalent 5.2.15
,
,
SL(n, IR) be the stabilizer of a point in IP" - 1 and Nn its maximal normal amenable subgroup. Then Hn/Nn is locally isomorphic to PSL(n - 1 , IR) If n, p � 3, and n # p, it follows from Theorem 5.2.14 (using the above remark for essential freeness) that the actions of SL(n, Z) on IP" - 1 and SL(p, Z) on 1Pr 1 are non-orbit equivalent However, for n = 2, the action of SL(2, Z ) on IP 1 is amenable, while for n � 3, the action of SL(n, Z) on IP" - 1 is not amenable (Example 4.3.8).. From Proposition 4.JJO, we deduce that SL(2, Z ) acting on IP 1 is not orbit equivalent to SL(n, Z) acting on lP" - 1 , n � 3, and this completes Proof: Let Hn
c
the proof of (a).. Assertion (b), (c) are proved similarly. Finally, in concluding this section, we mention without proof one further result on orbit equivalence which can be obtained from the techniques developed here. If H is a connected Lie group, N the maximal normal amenable subgroup, and S an ergodic H-space, the action is called irreducible if the restrictio n of the action to any normal subgroup of H which properly contains N is still ergodic. If H is semisimple, with no compact factors, this reduces to our previous definition (2.2J 1 ), and if H/N is simple, then this is no further restriction than ergodicity of H
113
Rigidity
[Zimmer 9]: For i = 1 , 2 , let H i be a connected Lie group with maximal normal amenable subgroup Ni Suppose IR-rank (H J / N t ) ;:;:; 2. Let Si be an essentially free irreducible ergodic Hi space with .fi nite invariant measure, and suppose the H 1 action on S 1 and the H 2 action on S 2 are orbit equivalent Then H d N 1 and H 2 / N 2 are isomorphic and N 1 is compact if and only if N 2 is compact . 5.2.1 6
Theorem
For implications of these results for foliations, see [Zimmer 8]. For some further results on orbit equivalence, see chapters 9 and 1 0..
6
6.1
Margulis' ArithmeticityTheorems Arithmeticity in groups of iR-rank � 2
We recall from the introduction the following construction of lattices . ( 1 ) If G is a connected semisimple algebraic CQ-group, then Gz is a lattice in Grr1 (Theorem of Borel-Harish-Chandra, 3. L7) (2) If G is locally compact, r c G a lattice and r' c G a subgroup commensurable with r, then r' is a lattice . (We recall that r, r' commensurable means [r: r n r'], [r' : r n r'] < oo ..) (3) If r c H is a lattice and cp : H -+ G is a surjective homomorphism with compact kernel, then cp(r) is a lattice in G. (To see this, we first observe that since kernel(cp) is compact, cp(r) is discrete . Furthermore, there is a natural H-map of H-spaces cp : Hjr -+ Gjcp(r), and hence if J1 is the H-invariant probability measure on Hjr, cpAJl) will be a G-invariant probability measure on Gjcp(r)) A lattice in a semisimple Lie group is called arithmetic if it can be obtained from the above three procedures. More precisely:
G be a connected semisimple Lie group with trivial center and no compact factors Let r c G be a lattice. Then r is called arithmetic if
6.1.1
Definition: Let
there exist (i) a semisimple algebraic CQ-group H and (ii) a surjective homomorphism cp : H � -+ G with compact kernel, such that cp(Hz n H �) and r are commensurable.
We remark that we may assume that the center of H � is trivial (simply by dividing it out if it were present) and by semisimplicity, H � is isomorphic to G x K where K is a compact group, and cp is simply projection onto G. Our main aim in this section is to prove Margulis' arithmeticity theorem.
[Margulis 1]: Let G be a connected semisimple Lie group with trivial center and no compactfactors. Let r c G be an irreducible lattice. Assume IR-rank(G) :;:;; 2. Then r is arithmetic. 6.1.2
Theorem
Margulis' arithmeticity theorems
115
Before beginning the proof, we shall discuss a very useful operation on algebraic groups, namely "restriction of scalars." We do this both because it will be used in the proof of Theorem 6.1 . 2 and because it provides a basic construction of algebraic �-groups (and hence of arithmetic groups) Suppose k is an algebraic number field, i e , � c k c e and d = [k, �] < oo Suppose G is an algebraic k-group (or more generally, an affine k-variety) . We wish to describe the construction of an algebraic �-group H such that Gk and H� are naturally isomorphic. For example, we can view e N as a group defined over k, and the k-points are of course kN Choosing a basis of k over �. we can identify k N and � Na, and � Nd can of course be considered as the �-points of the �-group e Nd Thus, by expanding e N to e Nd' the k-points become �-points . We will now describe this in a more formal fashion which will make clear the extension of this notion to arbitrary groups (or varieties). Let d = [k : �] < oo, k c IL, and choose a basis a 1 , . , a a of k over �. We recall that there are d distinct field embeddings 0" 1 , . , O"a , O"i : k -+ e, and we choose 0" 1 = identity. It is well known that: (i) { O"i} are linearly independent in the space of functions k --+ e (which is a vector space over C), and hence the d x d matrix (D"i(a i) ) is non singular; and (ii) If o: E k, then O"i(a) = a for all i if and only if IX E � Now fix a positive integer N We wish t o describe the identification kN � � Nd as an identification of k-points in e N with �-points in e Nd Each O"i defines a map O"f : k N --+ e N; we shall abuse notation and still denote this map by O"i : kN --+ eN For each w E kN, let w' E(e N )a be given by w' = (0" 1 (w), , O"a(w)). Then w -+ w' is a bijection of kN with its image, which we denote by (kN )' c (e N )a. If we let p : (e N )a -+ e N be projection onto the first factor, then p i (k N )' : (kN)' -+ kN is the inverse map to w --+ w'. These maps establish an isomorphism of kN and (kN )' as �-vector spaces. We identify (eN )d with d x N complex matrices . The group GL(d, C) acts on the set of d x N matrices by multiplication on the left, and thus we have an action of GL(d, C) on (e Nt Let TE GL(d, C) be the matrix T= (D"i(1Xi)) described above . Then we consider T as a map T: (e N )d -+ (eN )d It is immediate from the definitions that TI (� N )d is a bijection T: (� N )d --+ (kN )' (Formally, then T provides an isomorphism of (e N )d with a �-variety such that identifying (eN )d with this �-variety, [ (e N)a]� = (kN)') If we let (!) be the ring of algebraic integers in k, then we can choose the basis a 1 , , aa of k over � such that ai E (!) and (!) = LZai I n this case T will satisfy the additional condition that T((Z N )d) = ((!) N )' We also observe that if K is a field with O"i(k) c K c e for all i, then T((K N )d) = (K N )d Suppose that p E k[X 1 , . . , XN ] If O" : k --+ e is a field embedding, we let p" be the corresponding polynomial with coefficients in O"(k). We recall that O" can be
Ergodic theory and semisimple groups
116
extended to a field automorphism <J E Aut(C) Thus, if I c e [ X 1 , . . . , Xn ] is an ideal, we let !" = { pa l p EI}. Then !" is an ideal, and if I is generated by elements of k[X 1 . , Xn ], J<1 is generated by elements of
X
o
'
We have S(a 1 (p(x)), . , ad(p(x)))E(k N )', and hence upon applying T- 1 we obtain an element of (rp N)d Therefore, under the identification of (eN)a with (eN)a via T, this variety ll V"i is actually defined over rp. Furthermore, (ll V"i)(Q = ( Vk)' Where Vk iS the image Of Vk under the isomorphism kN -> (k N ) We summarize our discussion in the following. (We shall only be using this construction for groups, and hence just summarize for this case. ) '..
6.1.3 Proposition (Restriction of Scalars) [ Weil l] Let k be an algebraic number field ( [k:rp] = d), and suppose G is an algebraic k-group.. Let a 1 , . . . ,
the distinct field embeddings of k into e with a 1
d
identity. Let Rkt(Q(G) = fl G"', i= 1 and for g E Gk let g' = (a 1 (g), . , aa(g)). Let (Gk)' = {g'l g E Gk } . Then Rkt(Q (G) is (isomorphic to) an algebraic rp-group such that (Rkt(Q (G))(Q = (Gk)', and (Rk;(Q (G))z = (GI'!)' where (!) c k is the subring of algebraic integers in k. The projection map p:Rkt(Q(G) -> G onto the .first factor is defined over k, and defines bijections (Rk/(Q (G))(Q -> Gk, (Rk/(Q(G))z -> GI'! Furthermore, if for all i, a i (k) c K where K is a field k c K c e, then each Ga, is defined over K and (Rk/(Q(G))K = ll (Ga'k The group Rk/(Q(G) is called the restriction of G to rp. Theorem 3.1.7 implies: =
1 17
Margulis' arithmeticity theorems
Corollary: If k is an algebraic number field, and G is a semisimple algebraic k-group, then Gl!l is isomorphic to a lattice in [Rk;
6.1.4
6.1.5
Example: We consider in this light Example 5..21 2.. Namely, let
G = O (m, n) c GL(m, C) be the subgroup leaving the form
m-n
I
i= 1
xf - }2
m
I
i=m-n+ 1
xf
invariant Thus, G is an algebraic group defined over k =
m
"
i:l xf + j2 i = mI- n + l xf invariant and Rk;
i=
G x G " Since k, () (k) c IR, ( G x G")IR = GIR x (G")IR . By Corollary 6 . 14 (GI!I)' is a lattice in (G x G")IR, and since (G")IR is compact, the projection of G� to GIR is a lattice . Thus, Gzr�21 n G� is an arithmetic lattice in G� With these preliminaries, we now turn to the proof of the arithmeticity theorem. As the reader will see, the basic part of the argument is Margulis' superrigidity theorem which we proved in section 5.. L During the course of the proof, we will need the fact that r is finitely generated. If all simple factors of G have IR-rank at least 2, this will be proven in Chapter 7. (See Theorems 7.. 1 4, 7 . 15 .) For the result in general, see [Raghunathan l J .
Proof of Theorem 6.1.2. We shall change our notation and let G be a connected semisimple algebraic
lattice. We assume IR-rank (G) � 2 and G � has no compact factors . (We can assume we are in this situation by Proposition 3.. 1 . 6..) If G c GL(n, C), let L (G) be the Lie algebra of G, i e. , L(G) is the tangent space at the identity, which we can consider as a linear subspace of M (n x n, C), the space of all n x n complex matrices . Since G is a
Lemma:
For all y E r, Tr(Ad(y)) is a (real) algebraic number. (Here Tr
denotes trace.) Proof: Since the group of field automorphisms Aut (C) acts transitively on the
1 18
Ergodic theory and semisimp1e groups
transcendental numbers, it suffices to show that for each y E !, {O"( Tr (Ad(y) ) )l O" E Aut ( C) } is a bounded set of complex numbers. Each O" E Aut(C) acts on M (n x n, 0::: ) by (a;i) -> (O"(a;i) ), and since G and L(G) are defined over
=
I Tr (Adn, (p;(O"(y) ) ) ).. i
Since 1 is Zariski dense in G by the Bore! density theorem (3..2.5), O" (l) clearly is as well, and hence (p; o O") (l) is Zariski dense in H; . Thus, by Margulis super rigidity (5 . L2), either (p; o O")(l) is contained in a compact subgroup of H;, or (p; o 0")11 extends to a rational surjective homomorphism n:G -> H;. In the for mer case, all eigenvalues of Adn,(p;(O" (y) ) ) ) have absolute value 1, and hence I Tr (Adn, (p;(O" (y) ) ) )l � dim H;. If p; o O" extends to n, then letting Cg : G -. G and ch:H; -> H; be conjugation by g E G and h E H;, n o Cg = c"(gJ a n, and hence upon differentiating, we obtain for each g E G the commutative diagram L(G) dn
Ad(g)
1
1
L (H;)
L(G)
Adn.(n(g))
dn
L(H;)
Since dn is surjective, any eigenvalue of Adn, (n(g)) is an eigenvalue of Ad (g), and in particular, for y E 1, any eigenvalue of Adn, (p;(O"(y) ) ) is an eigenvalue of A d (y).. Thus, if we let e (y) be the maximal absolute value of an eigenvalue of Ad (y), we deduce that for any O" E Aut (C), I Tr (Ad(O" (y) ) ) l � (1 + e (y) ) dim G. This proves the lemma The next step is to pass from the conclusion that the traces are algebraic to the assertion that we can assume that the matrix entries themselves are algebr aic.
Let K be the field of real algebraic numbers. Then for some m, there exists a faithful IR-rational representation n:G -> GL(m, q such that 6.1 .7
Lemma:
Margulis' arithmeticity theorems
1 19
n(r) c GL(m, K ). Thus, identifying G with n(G), we can assume that G is defined over K and r c G K Proof: Let T:G ---> C be given by T(g) = Tr (Ad (g) ). This is a polynomial function, and hence the linear span V of { g Tlg E G} is finite dimensional, where g acts by translation on polynomial functions, i e , (g T)(h) = T(hg). Let n be this representation of G on V Assume for the moment that n is faithful. Since r is Zariski dense in G, V is also spanned by { n(y) Tly E r}, (for otherwise we would have a r-invariant subspace under a rational representation of G which was not G-invariant) Choose y 1 , , Ym E r such that { n (y;) T} is a basis for V It suffices to show that with respect to this basis the matrices of n (y) have all entries in K, for all y E r. Since elements of V are polynomial functions, and { n(y;) T} are linearly independent as functions on G, the Zariski density of r in G implies that they are linearly independent as functions on r. Therefore, we can find s 1 , . , Sm E r such that the matrix A = (n(y;) T)(si)) = (T(sm)) is non singular.. By Lemma 6.1 . 6, A E M (m x m, K) . Let C (y) = (c;i(Y)) be the matrix of n (y) with respect to { n(y;) T} . Thus, n (y)(n (y;) T) = L cii(y)(n (yi) T).. Evaluating
j L c;i(Y) T(skyi). Lemma 6.. 1.6 implies j that T(skyy;) E K, so for some matrix B E M (m x m, K), we have B = C(y) A. Since A E M (m x m, K) as well, and A is invertible, we deduce C(y) has entries this equation at sk, we obtain T(skyy;)
=
in K. Thus, to verify lemma 6.. 17 it suffices to show that the representation n is faithfuL Suppose g E ker(n).. Then g T = T, i.e. Tr(Ad(hg)) = Tr(Ad(h)) for all h E G. Let W be the linear span of {Ad(h) I h E G } in the space of all endo morphisms of the linear space L(G). We then have Tr(MAd(g)) = Tr(M) for all M E W Since G is semisimple, Ad is a direct sum of irreducible representations. Thus we can write the representation (Ad, L(G)) � LEB(O";j, V;i) where each O";j is irreducible, (O"; j , V;i) = (O";k, V;k), and for i i= r, O"ik and O",k are inequivalent Any endomorphism of LEil V;i leaving each V;i invariant is given by a family M;i E End( V;i). From the irreducibility of O";i it follows (by Wedderburn's theorem) that under the above identification L(G) � LEB V;h we have
W = { M E End(LEil V;i) I M( V;i) c V;i all i, j, and M;i = M;k for all i, j, k } . From the fact that Tr(MAd(g)) = Tr(M) for all M E W, we then deduce that Tr(SO";/g)) = Tr(S) for all S E End( V;i)· It follows that O";/g) = I for all i, j, and hence that Ad(g) = I . Since G has trivial center, g = e completing the proof We will now use the fact that the group r is finitely generated . We will prove
120
Ergodic theory and semisimple groups
this in the case in which all simple factors Of G have IR-rank at least two in chapter 7 . (See 7. 1.5 . )
6.1.8 Lemma: With G as in the conclusion of 6. 1 . 7, there is a real algebraic number field k ( [k Q] < oo) such that G is de,fined over k and f c Gk Proof: If { yi} is a finite set of generators of f, let k be the field generated by the matrix entries of Y i Then by 61 7, [k : Q] < oo, and clearly r c GL(m, k). Since f is Zariski dense in G, it follows from Proposition 3. L8 that G is defined over k
We now complete the proof of the theorem. Let Rk;CQ(G)
=
flG""; be as in i
Proposition 6. 1 3.. Let a : Gk --> Rk;CQ(G) be the map a(g) = (
Margulis' arithmeticity theorems
121
of the matrix entries of a(y) E Hro are uniformly bounded over y E r. More formally, if we let Za c (Q;, be the a-adic integers, then Hz" c Hroa is open, and hence boundedness of a(r) implies that the image of a(r) in Hro)Hz" is finite. l is finitely generated, and hence there are finitely many primes a 1 , . , an such that any denominator of any matrix entry of a(r) E Hro is a product of powers of the a; In other words, the map n
a(f')/ a(f) n Hz --> TI H'Oa ' / HZa i= 1
'
is mJective Thus, from the finiteness of a(f') in Hro)Hz", we deduce that a(r) n Hz is of finite index in a(r) Applying p, we deduce that r n p(Hz) is of finite index in r . Since r is a lattice in GR, this implies that r n p(Hz) is also a lattice . By assertion 3 at the beginning of this chapter, p(Hz) is a lattice in GR since (ker P)R is compact Since er n p(Hz)) c p(Hz) is an inclusion oflattices, this implies that f' n p(Hz) is of finite index in p(Hz). Thus, r and p(Hz) are commensurable (and hence so are r and p(Hz n H 2)). This completes the proof of Theorem 6.. 1 . 2 With notation as in Theorem 6. 1 . 2, and Definition 6.. 1.1 , if r is a non-eo compact lattice, the map p : H g --> G can in fact be taken to have not only compact kernel, but trivial kernel, i.e. , p can be chosen to be an isomorphism. The proof of this from Theorem 6.. 17 depends on the following compactness criterion, which should be considered as a companion result to Theorem 31 ..7. 6.1.9 Theorem [Borel�Harish�Chandra 1 ] , [Mostow�Tamagawa 1 ] : Let G be a semisimple algebraic (Q-group. Then GR/Gz (which is of finite volume by 3 . 1.7) is compact if and only if the only unipotent element in Gro is the identity.
We then deduce Corollary [Margulis 1]: If G is a connected semisimple algebraic IR-group with trivial center and IR-rank(G) � 2, and r c cg is an irreducible non cocompact lattice, then there is a connected semisimple algebraic (Q-group H with trivial center and an isomorphism p · HR --> GR such that p(Hz) and r are commensurable. 6.1.10
Proof: By Theorem 6. L2, we can find a connected semisimple algebraic
(Q-group H with trivial center and an IR-rational surjective homomorphism p : H --> G such that p(Hz) and r are commensurable and (ker p)R is compact. If r is not cocompact in cg, then Hz is not cocompact in HR Thus, by
Ergodic theory and semisimple groups
122
Theorem 6.. 19, H'f:i possesses non-trivial unipotent elements. Since ker p is a normal IR-subgroup, there is a normal IR-subgroup L c H such that the product map L x ker p � H is an isomorphism of IR-groups. If we let q : H � ker p be the projection map then q is IR-rational, and hence if x E HIf:! is unipotent, so is q(x) E (ker p)11. Since (ker p)111 is compact, q(x) = e, and hence H'f:i n L i= {e} Since L c H is normal, H IQ n L is normal in HIf:! , and since HIQ c H is Zariski dense (by 3 1 . 9) the Zariski closure of H IQ n L, say L 1 . is a normal subgroup of H. Since L 1 contains (by definition) a Zariski dense set of �-points, L 1 is a �-group by Proposition 1 L8.. Thus, (L 1 )z is a lattice in (L 1 )111 by 31 .7, and since p : L � G is an isomorphism, p( (Ll )z) is a lattice in p(L 1 )111 and in particular is infinite. Since p(Hz) and r are commensurable, this implies r n p(L 1 )111 is infinite. Because r is an irreducible lattice in Gg, this implies that p(Ll) = G. (This follows from the general fact that if G
=
n
fl G i is a product of connected simple
i= 1 Lie groups with trivial center and r c G is an irreducible lattice, then m
r n TI G i = { e} as long as m < n. This is so because r0 is normalized i= 1 by r and TI Gi, and hence r 0 is normal in G since r TI G i , is dense r0
=
i>m
i>m
b y the irreducibility assumption. Since G i are center free and r 0 i s discrete, this is impossible unless r0 is triviaL) Thus, L1 = L, and hence L is actually defined over � · Thus Lz is a lattice in L111 , and since (ker p)111 is compact, Lz is a lattice in H 111 Thus Lz c Hz is a subgroup of finite index, and p(Lz) and r are commensur able. Thus, replacing H by L, we obtain the required �-group.
6.2
The commensurability criterion
In this section we present a result of Margulis which gives a necessary and sufficient condition for a lattice to be arithmetic without the assumption that the IR-rank of the ambient Lie group is at least two .
Definition: Let G be a locally compact group and r c G a closed subgroup. Let CommG(r) = {g E G i grg - 1 and r are commensurable } . Then CommG(r) is called the commensurability subgroup of r in G If G is understood, we shall sometimes denote this simply by Comm(r) 6.2.1
..
We remark that Comm(r) is a subgroup of G and r c Comm(r) c G. If r, r' c G with r and r' commensurable, then Comm(r) = Comm(r'). As an example, we have the following.
Margulis' arithmeticity theorems
123
[Borel 5]: Let G be a connected semisimple algebraic �-group with trivial center. Then G� c Comm(Gz) and if G" has no compact factors, then G� = Comm(Gz).
6.2.2
Proposition
Proof: Let g E G�, and let m be the least common multiple of all denominators of the entries of the matrices g, g - l Let r = { y E Gz I y = I mod m 2 }, so that r c Gz is clearly of finite index. If y E r, then y = I + m 2 B where B is an integral matrix, and hence gyg - 1 = I + m 2 gBg- 1 which is also an integral matrix . Thus grg - 1 c Gz Furthermore, since r is of finite index in Gz, r is a lattice in G" and hence so is g rg 1 . Therefore, g rg - 1 must be of finite index in Gz Since g rg - 1 is clearly of finite index in gGzg- 1 , it follows that g E Comm( Gz) To see the converse, let C [G] and �[G] denote as usual the space of regular functions on G and the space of �-regular functions on G respectively.. Let V be the subspace of C[G] spanned by the matrix coefficient functions, i e , g ---> g;h and V� the subspace of �[G] spanned by these functions. Then for some m, we can identify V � c m in such a way that V� � � m Define a representation of G on C[G] by (n(g)f)(x) = f(g - 1 xg) . Then Vis G-invariant, and n : G ---> GL( V) is a rational representation defined over �. Since G has trivial center, n is faithful, and thus n is a �-isomorphism of G with n(G). Thus, if g E G, to see that g E G� it suffices to see that n(g)fE V� for all fE V� Suppose that g E Comm(Gz) and [E V� Then we can write n( g)f = -
.
.
n
fo + L: c;f;, where fi E V�, c; E C and (1, c 1 , . . , c.) are linearly independent over i= 1 �·· Since g E Comm(Gz), gGzg- 1 n Gz is a lattice in Gp, and hence by the Borel density theorem (.1 2.5), gGzg - l n Gz is Zariski dense in G. If x E g Gzg - l n Gz, then f(g - 1 xg) = fo(x) + L;cdi(x) where f(g - 1 xg), fo(x) E � By linear independ ence of { 1 , c;} over �, we deduce that fi(x) = 0, f(g - 1 xg) = f0(x) for all x in a Zariski dense set Thus, this holds for all x E G, and hence n(g)f E V� By the conclusion of the preceding paragraph, g E G�, and this completes the proof. .
For an irreducible lattice, we have the following dichotomy.
Let G be a connected semisimple Lie group with trivial center and no compact factors, and let r c G be an irreducible lattice.. Then either r c:: Comm(r) is a subgroup of finite index or Comm(r) is dense in G (with the Hausdorff topology). 6.2.3
Proposition:
= Comm(r) and H 0 the (topological) connected component of the identity. Since r c H, r normalizes H 0 , and hence L(H 0 ), the Lie algebra of Proof: Let H
124
Ergodic theory and semisimple groups
H 0 , is an invariant subspace of L(G) under Ad(r). Since r is Zariski dense in G, L(H 0) is invariant under Ad(G), and hence G normalizes H 0 If H 0 = {e} , then H and hence Comm(r) are discrete, and since r c Comm(r) c G and r is a lattice in G, [Comm(r): r] < oo . On the other hand, if H 0 is a non-trivial normal subgroup then rH 0 is dense in G by irreducibility of r, and since rH 0 c H, G = H, completing the proof For G = SL(n, IR), and r = SL(n, Z), it is clear from Proposition 6.2 2 that Comm(r) is dense in G with the Hausdorff topology. The next proposition asserts that this is true in a more general setting Proposition: Suppose G, r are as in Proposition 6. . 2 . 3 . If r is arithmetic, then Comm(r) is dense in G .
6.2.4
Proof: Let H be a connected semisimple CQ-group and p : H g --+ G a surjective homomorphism with compact kernel such that p(Hz n H 2 ) is commensurable with r. By 6. 2.3 it suffices to show that Comm(r) is not discrete. It suffices to see that p(CommHrFlHz)) is not discrete, and hence by 6.2. 2 that p(Hrq) is not discrete. To establish this, we call on a result of [Bore! 6] described in Chapter 10.. Namely, if a E Z is a prime number, let Z(a) c CQ be the subring consisting of all rational numbers whose denominator is a power of a. We have natural non discrete embeddings of Z(a) into IR and COa, where COa is the field of a-adic numbers. However, the image of Z(a) in IR x COa will be discrete and in fact will be a lattice in IR x COa · [Bore! 6] establishes that the analogous result is true for H. Namely, under the natural embedding A : Hz
induced by projection onto H� is easily seen to be injective, and hence it suf fices to see that KA(Hz
Margulis' arithmeticity theorems
125
we have an Hr? x Hq;,a-map (Hr? x Hq;,a)/K.A.(Hz) -+ Hq;,)Hza· Therefore, H0) Hza would have a finite Hq;,a-invariant measure . Since Hza is compact and H Oa is not compact, this is impossible. The main result of this section is the converse to Proposition 6 ..2.4. Theorem [Margulis 1]: Let G be a connected semisimple Lie group wi'th trivial cent er, no compact factors, and let r c G be an irreducible lattice. If Comm(r) is dense in G then r is arithmetic.
6.2.5
Thus, the dichotomy in Proposition 6.. 2 .. 3 is exactly the dichotomy of non arithmeticity and arithmeticity. To begin the proof, we first prove a replacement for the superrigidity theorem (5. L2). The conclusions of the following theorem are exactly those of 5. 1 . 2, but the assumption in 51.2 that the IR-rank be at least two is replaced by the assumption that the homomorphism of r extends to a homomorphism of Comm(f).
[Margulis 1]: Let G be a connected semisimple algebraic IR-group such that G� has no compact factors. Let r c G � be an irreducible lattice, and assume that Commcii<(f) is dense in G� Suppose k is a local field of character istic 0, and let H be a connected algebraic k-group, almost simple over k. Let n : Comm(r) -+ Hk be a homomorphism with n(Comm f) Zariski dense. Then
6.2.6
Theorem
(i) If k = IR, HR is IR-simple and non-compact, then n extends to an IR-rational homomorphism G -+ H (ii) If k = C and H is simple (i. e. center free), then either (a) n(f) is compact or (b) n extends to a rational homomorphism G -+ H (iii) If k is totally disconnected, n(f) is compact Proof: Lemma 5. L3 and Proposition 5 . 1 . 9 show that it suffices to prove the
following two assertions (cL Steps 1 , 1 ', 2 in the proof of 5. 1 .2 )
k = IR, it must be (a) ): (a) There is a proper algebraic k-subgroup L c H and a measurable Comm(r)-map cp : G �/Po -� Hk/Lk; or (b) There is a compact subgroup S c Hk and a measurable f-map cp : G �/ Po -+ Hk/S (2) In case (la), for k = IR or C, cp is essentially rational, and for k totally disconnected, cp is essentially constant ( 1 ) One of the following must be satisfied (and for
Ergodic theory and semisimple groups
1 26
To deduce the theorem from assertions ( 1), (2), we observe that the proof of Lemma 51 3 and Proposition 5. L9 immediately take care of k = IR, C For k #- IR, IC, Proposition 5 . 19 shows that it suffices to consider case ( l a).. Just as in the conclusion of the proof of Theorem 5. L2, if we had ( l a), by (2) cp would be essentially constant, and this would imply that n(Comm(r)) c Mk where M is a proper algebraic k-subgroup of H (in fact a conjugate of L). This contradicts the assumption that n(Comm(r) ) is Zariski dense in H This verifies that it suffices to prove assertions (1) and (2). Assertion ( 1 ) is of course the analogue in the present context of Step 1 (for k = IR) and Step 1 ' (for k #- IR) of the proof of 5. 1.2 . We remind the reader that the assumption that IR-rank(G) � 2 is not used in the proof of Steps 1 , 1 ' i n 5. 1 .2.. Therefore, we still have the assertions o f Steps 1 , 1', yielding the existence of suitable r-maps. In our present case (la), we need to obtain a Comm(r)-map. Therefore, assertion ( 1 ) follows from Steps 1 , 1 ' of the proof of Theorem 5 . 1 . 2 and the following result
Lemma: Suppose cp G �/Po ----+ Hk/Lk is a measurable r-map where L c H is a proper algebraic k-subgroup. Assume further that L is of minimal dimension among all k-subgroups for which such a measurable r -map exists. Let N(L0 ) be the normalizer of L0 , and q Hk/Lk ---. Hk/N(L0 )k be the natural projection.. Then q o cp · G �/Po ----+ Hk/N(L0 )k is a Comm(r)-map 6.2.7
·
To prove Lemma 6. 2.7, we first show the following. Lemma: Assume the notation of 6..2. .7. Let r 0 c r be a subgroup offinite index.. Suppose t/1 . G�/ Po ----+ Hk/Lk is a measurable r0 -map. Then q tjJ = q o cp (a.e.).
6.2.8
o
Proof: Recall the r-action on the space of measurable maps F(G�/P0, Hk/Lk) given by (y f)(x) = f(xy)y - 1 Thus /E F(G�/P0, Hk/Lk) is (essentially) a r-map if and only if f is a fixed point under r. Since t/J is fixed under r o, t/1 has a finite orbit under the r-action, say r t/1 = { t/1, Y 1 · t/J, . . , Yn t/1 }. Let S(n + 1 ) be the . .
symmetric group on n + 1 letters, which acts naturally on n
A : G �/Po ---+
0 Hk/Lk
i=O
by A(x)
=
n
0 Hk/ Lk.
i=O
(t/J(x), (y l · t/J)(x), . . . , (Yn ' t/i)(x) ),
and let X be the composition o f A with the projection
Define
127
Margulis' arithmeticity theorems
Thus, X is a r-map. We therefore also have a r-map
(cp, X) : G�/Po --+ Hk/Lk The action o f Hk o n Hk/ Lk
x
(Do Hk/Lk );s(n + (a );s(n + x
Hk/ Lk
1)
1 ) i s smooth . This follows from
3. 1.3. (More precisely, smoothness of this Hk action is equivalent to smoothness
of the Hk
X
S(n + 1 ) action on Hk/Lk
X
n
n Hk/Lk
Hk is smooth on this space
i=O
by 3.. L3, and hence so is Hk x S(n + 1 ), by Lemma 3.2. 1 4.) Thus, the space of Hk-orbits is countably separated, and (cp, X)', the composition of (cp, X) with projection to the space of Hk-orbits will be essentially constant, i.e. , there is a single Hk-orbit such that (cp, X)(x) lies in this orbit for almost all x. We can choose a point in this orbit whose stabilizer will be Lk n M k, where M is the subgroup of H leaving some finite subset of Hk/Lk invariant. We can consider (cp, X) as a r-map (cp, X) : G�/Po --+ Hk/Lk n Mk. However, a subgroup of finite
n
index in M k leaves a finite set pointwise fixed, and so is of the form n a;Lka;- 1
(a Hk/Lk);s(n + i=O
for some a; E Hk, where ([e], { [ao], . , [an ] } ) E Hk/Lk
x
1)
lies in the orbit containing almost all (cp, X)(G�/P0 ). Since (cp, X) is a r-map, the minimality of dimension hypotheses on the group L implies dim(M n L) = dim L, and hence dim((n aiLaj 1 ) n L) = dim L. This implies that for each i, a;L0 a;- 1 = L 0 , i.e. , a; E N(L0)k. Thus the image of ( [e] , { [a0 ], . . . , [an ] }) in Hk/N(L0 )k x (ITHk/N(L0 )k)/S(n + 1 ) lies in the diagonal and (since the diagonal is Hk-invariant), so does the entire orbit of ([e], { [a0 ], . . , [an] } ) However, this contains almost all (q(cp(x)), (flq)(X(x))), and hence for almost all x E G 2/ P 0 , (q(cp(x)), q(I/J(x))) lies in the diagonal in Hk/N(L0 )k x Hk/N(L0 )k. This completes the proof of Lemma 6 ..2..8. ..
Proof of Lemma 6.2.7. The action we defined at the beginning of the proof of
Lemma 6..2.8 of course extends to an action of Comm(r).. We have that cp is fixed under r. Let g E Comm(r). Then there is a subgroup r 0 c r of finite index such that g - 1 r0 g c r. It follows that g · cp is fixed under r 0 By Lemma 6.2.8, q a (g · cp) = q o cp . But q o (g · cp) = g · (q o cp), where the Comm(r) action on F(G2/P 0 , Hk/N(L0 )k) is defined similarly.. Thus, for all g E Comm(r), g (q cp) = o
Ergodic theory and semisimple groups
128
q cp, i e , q o cp is a Comm(r)-map. (More precisely q o cp is essentially a o
Comm(f)-map, and hence it agrees almost everywhere with a Comm(r)-map . )
This completes the proof of assertion ( 1 ) above, and hence to prove theorem 6 2. 6 it suffices to verify assertion (2) The argument will be similar to that of the proof of Step 2 of theorem 5 1 . 2, although it is in fact a bit simpler. We lift cp to a map cp : G� -+ Hk/ Lk . and cp is a Comm(r)-map. As in Lemma 5. 1 . 5, it suffices to show that for some g E G�, u -+ cp(ug) is an essentially rational function of u E U R if k = IR, C, or an essentially constant function for k # IR, C For g E G�, let cp9 : G� -+ Hk/ Lk be given by cpg{a) = cp(ag).. Then the proof of Lemma 5 1 . 8 shows that it suffices to show that for almost all g E G � there exist: (a) (b) (c) (d)
a k-subvariety W9 c H/L such that cp9(a) E W9 for almost all a E G�; an algebraic k-group Q9 which acts k-regularly on W9: a measurable homomorphism h 9 : G� -+ (Q9)k; a point x9 E W9 n Hk/Lk such that for almost all a E G�, cp9(a) = x9 hg(a).
To see the existence of (aHd), it suffices by Proposition 352 to see that for almost all g E G�, cp9 all lie in a single Hk-orbit in F(G �, Hk/ Lk) Let : G � -+ F( G �, Hk/Lk) be ( g) = cp9 If h E Comm(r), then for each c and almost all g, we have cp9h(c) = cp(cgh) = cp(cg) ·· n(h) = cp9(c)n(h). By Fubini, for each h and almost all g, this is true for almost all c. Thus, for each h E Comm(r), we have
Corollary: With notation as in Theorem 6.2. 6, assume f c Comm o (r) Grr1 is a subgroup offinite index with r c f. Assume n f -+ Hk is a homomorphism with n(f) Zariski dense. Then the conclusion of Theorem 6.2.6 is still true.
6.2.9
Proof: The proof of Theorem 6.2.6 will apply verbatim with Comm(r) replaced
by f once we know f is dense in G� But if A is the closure of f in G�,
Margulis' arithmeticity theorems
129
since Comm(l)/f is finite, finitely many cosets of A will contain Comm(i) and hence G2, since Comm(i) is dense in G2 Since G 2 is connected, A = G2 We now return to the proof of the arithmeticity criterion
Proof of Theorem 6.2.5. Once again, we change our notation from the statement
of the theorem, and assume r is an irreducible lattice in G� where G is a connected semisimple algebraic CQ-group with trivial center and G2 has no compact factors In the proof of Theorem 6. 1.2 (arithmeticity for irreducible lattices in IR-rank at least 2), the super rigidity theorem (51 .2) was applied only to certain homomorphisms of 1 . The proof of Theorem 6.. 2.5 will be essentially the same as that of 61 . 2, except that we must verify that the homomorphisms of r in question actually extend to Comm(i) in order to be able to apply our substitute for superrigidity, namely Theorem 6.2 . 6. If CJ is an automorphism of C, then we obtain an induced map CJ : G ---> G (since G is defined over CQ). The homomorphism CJ I r obviously extends to Comm(i), and hence the proof of Lemma 6 . 16 shows that Tr(Ad(y)) is a real algebraic number for all y E r. Lemmas 61 7 and 6. 1 . 8 continue to be valid, so we may assume G is a k-group, r c Gk, where k is a real algebraic number field ([k : CQJ < ex:). We now claim that 1 c Gk implies that Comm(l) c Gk Let g E Comm(l). Then there is a subgroup 1 0 c 1 of finite index with gi 0 g - 1 c 1 .. Since r 0 is also a lattice in G2, 10 is Zariski dense in G, and hence (Proposition 3 . Ll 0) the inner automorphism Int(g) : G ---> G is defined over k. Thus Ad(g) : L(G) ---> L(G) is defined over k, i..e. , Ad(g) E Ad(G)k However Ad : G ---> Ad( G) is an isomorphism defined over k, hence g E Gk It follows that the homomorphism a : 1 ---> (Rk;(Q(G))(Q, o:(y) = (CJ 1 (y), .. , CJd (y)), as in the proof of 6. L2, extends to a homomorphism a : Comm(l) ---> (Rk;(Q(G))(Q Thus, letting H be the Zariski closure of o:(Comm(f)), and dividing by the radical and then the center as in 6. 1 2, we obtain a semisimple algebraic CQ-group H, an IR-rational epimorphism p : H ---> G and a homomorphism a : Comm(l) ---> H(Q such that o:(Comm 1) is Zariski dense and p o a = identity By replacing r by a subgroup of finite index and Comm(l) by f, a subgroup of Comm(l) of finite index, r c f c Comm(l), we can assume H is connected . Using Corollary 6. 2. 9 of (the proof of) Theorem 62. 6, we can complete the proof of Theorem 6 ..2.5 exactly as in the completion of the proof ofTheorem 6. 1 . 2
7
7.1
Kazhdan's Property (T)
Kazhdan's property and some consequences
Let G be a locally compact (second countable) group and n : G -+ U(£' ) a unitary representation of G on the (separable) Hilbert space £'. Of course, £' may or may not contain non-trivial vectors invariant under n( G ). We wish to define a notion of n almost containing invariant vectors
7.1.1
G a compact subset A unit vector v E £' is v 11 < c; for all g E K.. We say that n almost has
Definition: Let c; > 0 and K c
called (c:, K )-invariant if 11 n( g)v invariant vectors if for all (c:, K), there exists an (c:, K)-invariant unit vector (The vector may depend on (c:, K).) -
Example: Let n be the regular representation of lP on L 2 (1P).. Given c; > 0 and an interval [a, b] c lP, by choosing [c, d ] c lP to be a large enough 7.1.2
interval, and letting fr.c ,dJ E L2 (1P) be the normalized characteristic function of [c, d] , we will clearly have 11 n(t)j(,,dl - frc.dJ II < c; for all tE [a, b] Thus, n almost has invariant vectors. However, n does not have (non-trivial) invariant vectors The same argument shows that this is true for the regular representation of Z" or lP" . A basic discovery of [Kazhdan 1 ] is that many semisimple groups and their lattice subgroups do not have representations with this type of behavior. 7.1.3 Definition: G has the Kazhdan property (or "property ( T)") if any unitary representation of G which almost has invariant vectors actually has non trivial invariant vectors Thus, Example 71.2 shows that lP" and Z" do not have the Kazhdan property. However:
7.1.4 Theorem [Kazhdan 1]: Let G be a connected semisimple Lie group with finite center each of whose factors has !P-rank at least 2. Then G, as well as any lattice subgroup of G, has the Kazhdan property.
Actually, Kazhdan assumed that all simple factors had !P-rank at least 3
131
Kazhdan's property (T)
The fact that IR-rank at least 2 is sufficient was observed independently by [Delaroche-Kirillov 1], [Vaserstein 1], and [Wang 1J The proof of Theorem 7J A will be given in section 74. Kazhdan's property has proved useful in a large variety of situations, only a few of which we will be considering A basic consequence is the following
7.1.5
Theorem [Kazhdan 1]: Suppose r r is finitely generated
is a discrete group with the Kazhdan
property.. Then Proof: Let r
=
{yi} be a listing of the elements of r, and r" be the subgroup generated by {y 1 , . . , Yn} Let nn be the representation of r on L2(rjrn) given
by translation, i.e., the representation induced from the identity representation of r" Define n = � Ell nn The representation nn clearly contains a unit vector invariant under { y 1 , . . , Yn}, namely XUelJ· Thus, n almost has invariant vectors . Since r has the Kazhdan property, n has non-trivial invariant vectors. The projection of an invariant vector in �Ell L2(r;rn) onto each L2(r;rn) will be invariant, and for some n, will be non-trivial. Hence for some n, nn has non trivial invariant vectors, which clearly implies r;r" is finite. Since r" is finitely generated, it follows that r is as well. Another useful property is the following
is a (continuous) homomorphism with
Proposition: If
n is a representation of H which almost has invariant vectors, then n o
Proof: If
invariant 7.1.7
Corollary: If G
has the Kazhdan property, then every homomorphism into
IR" is trivial. This follows immediately from 7. 1 .6 and the sentence preceding Theorem 7.14. In fact, Corollary 7.1 .7 can be generalized considerably by using the following theorem.
7.1.8 Theorem [Hulanicki 1]: If G is a locally compact (second countable) group, then G is amenable if and only if the regular representation of G on L 2( G) almost has invariant vectors.
Ergodic theory and semisimple groups
132
The proof of this theorem will be given in section 7..2. We observe that the fact that amenability of G implies that the regular representation almost has invariant vectors is a generalization of Example 7 . 12. As a consequence of 7J 8, we have: 7.1.9
Corollary: If G
is amenable, then G has the Kazhdan property if and only
if G is compact. Proof: If G is amenable and has the Kazhdan property, then by Theorem 7 1 . 8
(and Definition 7 . 13), the regular representation has non-trivial invariant vectors. This implies that the Haar measure on G is finite, and hence G is compact Conversely, suppose G is compact and n almost has invariant vectors. If v is any unit vector, then iJ = JG(n(g)v)dg will be a G-invariant vector . Further, if 11 n(g)v - v 1 < t for all g E G, then 11 iJ - v 1 < ±, and iJ #- 0. Since n almost has invariant vectors and G is compact, we can choose such a vector v. We can then generalize Corollary 7. 1 7 as follows:
7.1.10
Let
cp
·
Corollary: Suppose G has the Kazhdan property and that H is amenable. G -> H be a homomorphism. Then cp(G) is compact.
Proof: cp(G) has the Kazhdan property by 7. . 16, is amenable by 4. 16, and hence
is compact by 7.1 . 9. 7.1.1 1
Corollary: Suppose r is a discrete group with the Kazhdan property.
Then
rj[r, r] is finite. This follows from 7. 1 . 10 and amenability of abelian groups (4 . 1.2). As another particular case of 7.. 1 . 10, we have (using 7 . 14): .
Corollary [Kazhdan 1 ]: Let G be a connected semisimple Lie group offinite center such that the IR-rank of every simple factor of G is at least two. Let r c G be a lattice and N c r a normal subgroup such that rjN is amenable. Then rjN is finite. 7.1.12
[Margulis 7] has shown that this last result is true if we only assume IR-rank(G ) � 2 provided that r is an irreducible. More precisely:
1 33
Kazhdan's property (T)
7.1.13 Theorem [Margulis 7]: Let G be a connected semisimple Lie group with no compact factors, finite center, and lR-rank(G) � 2. Let 1 c G be an irreducible lattice and N c r a normal infinite subgroup. Then rIN has the Kazhdan property, and in particular if r IN is amenable then rIN is finite.
We will not be proving 7. 1 1 3 here, and instead refer the reader to Margulis' paper. The remainder of this chapter is largely devoted to proving the assertions in this section which we have not yet verified: Theorems 7.. 14 (in section 74) and 71 . 8 (in section 7..2). 7.2
Amenability and unitary representations
In this section our aim is to prove that G is amenable if and only if the regular representation of G almost has invariant vectors . To this end, it will be convenient to introduce the notion of a mean on a function space. 7.2.1 Definition: Let (X, f.l.) be a standard measure space and F c L 00(X ) a (norm) closed subspace containing the constant functions, which is also closed under complex co11jugation.. A mean on F is a (norm) continuous linear func tional m : F � C such that
(a) m(1) = 1, where C is identified with the constant functions and hence with a subspace of L 00 (X }. (b) f � 0 implies m(f) � 0. (c) m(f) = m(f), or equivalently m(Re f) = Re(m(f)).
= {cp E L 1 (X ) I cp � 0, ll cp ll 1 = 1 }. Then m"'(f) = Jfcpdf.l. defines a mean m"' on L 00 (X), whenever cp E P(X ) 7.2.2
Example: Let P(X )
7.2.3 Proposition: (a) A F, the set of means on F, is a closed convex subspace of the unit ball in F * , where the latter has the weak- * -topology. (b) {m"'lcp E P(X ) } is weak- * dense in AF Proof: (a) is clear. To see (b), suppose the contrary. Then by Hahn-Banach, for some m E AF there exists e > O and /E F (recall that F � (F *) * , where F * has the weak- * -topology), such that Re(m(f)) � Re(m"'(f)) + e for all cp E P(X ). But this implies m(Re f ) > sup J (Re f)cp = ess sup (Re( f ) ). But this contra-
diets (b) in Definition 7..2. 1.
Ergodic theory and semisimple groups
1 34
If (X, /1) is a G-space, then G acts on L 00(X ) by isometries and hence, if L 00(X ) is a G-invariant subspace, G acts on A F However, while the action of G is continuous in G if L 00(X ) has the weak-*-topology as the dual of L t(X ), the action is not in general continuous on U)(G) if the latter has the norm topology, and hence we will not have continuity in general for the action on AF (Of course, if G is discrete, this is not an issue.) In any case, a fixed point in JltF will be called a G-invariant mean on F As an example of the usefulness of this notion, we have one direction of the equivalence we want F
c
Theorem: Suppose the regular representation of G almost has invariant vectors . Then G is amenable.
7.2.4
L 00(G). . Let n be the regular representation. By hypothesis, there is a sequence of unit vectors t/J i in L 2 (G) such that as j -+ oo, ll n(g)t/Ji - t/Ji li 2 -+ 0 uniformly on compact subsets of G . (This is essentially just a reformulation of Definition 7.. 1 . 1 .) Let cp i = I t/Ji 1 2 , so that ({Jj E P(G). An elementary argument then shows that 1 n(g)cpi - cpi li t -+ 0. (Namely, 11 n(g)t/J i - t/1 i 1 1 2 -+ 0 is equivalent to Re(j (n(g)t/J i)t/1 i) -+ 1 . Since I J(n(g)t/Ji)t/Ji l � J l n(g) t/Ji l lt/Ji l � 1 by the Schwartz inequality, it follows that J l n(g)t/Ji l l t/l i l -+ 1 also . Thus ll n(g) l t/Ji l - l t/JJ I I I 2 -+ 0 Hence Proof: We first claim that there is a G-invariant mean on
.
11 n(g) cpi - qJ i li t = J ( I n (g) t/1 i 1 2 - I t/1 i l 2 ) � J ( l n(g) l t/J i l - l t/li l l ) ( l n(g) l t/li l + l t/IJ I I ) � 11 n(g) I t/1 i I - I t/1 i I l l 2 1 1 n(g) I t/1 i I + I t/1 1 I ll 2 We have seen that the first term approaches 0, and the second is bounded by 2.) By Proposition 7 . 2 3 (a), A L'Xl
.
m(n(g) f - f) = lim m
lim Jf(xg)cpJx)dx - fh i
=
lim Jf(x)cp j(xg - t )dx - s r(/Jj
=
lim ff[n(g - t)cp i - cpiJ
Thus, l m(n(g)f -f) l � llfll oo lim ll n(g - t )cpi - cp i ll t = O. Hence, m is a G
i
invariant mean. To show amenability of G, let G act continuously on a compact metric space
1 35
Kazhdan's property (T)
X Fix a point x E X For /E C(X ), let lE Lco(G) be given by f(g) = f(xg). Define Jl(/ ) = m(l) Since m is a G-in variant mean, it follows that J1 E C(X ) * is a G-invariant positive linear functional, Jl(l ) = 1, and Jl( f) = Jl(f).. Thus, J1 is identified with a G-invariant probability measure on X, verifying amenability of G To prove the converse to Theorem 7.2 4 (i . e. Theorem 7.1 . 8), we would simply like to reverse the steps of the above argument This can be done in a straight forward fashion if G is discrete, but for a general locally compact group the argument will require some modification, although the basic idea is the same. We shall first present the proof for G discrete. Proof of Theorem 7.1.8 for G discrete: Suppose G is discrete and amenable. We first claim that there is a G-invariant mean on L 00 (G).. Since vi!L co (G) is a compact convex G-invariant subset of (Leo(G)) * , Proposition 4. 1.4 would guarantee the existence of a G-fixed point in vi{LOO(G) ifL 00(G) was separable in norm (which it is not in general).. However, since G is countable, we can write L 00(G) = u Fa where Fa c L eo( G) is a G-invariant separable closed subspace, closed under complex conjugation.. For each ex, we have the restriction map (L00(G ) )* � (Fa) * , which yields a map ra : ALco(G) � AF�·· By Proposition 4.1.4, there are fixed points in any AF�' as well as any AF� , + +F�" .. Let la = {m EALoo(G)Ira(m) is G-invariant} Since AF�, + +F�" has fixed points, {la} is a family of compact subsets of ALOO(Gh any finite subcollection having a non empty intersection. Since Ar oo(G) is compact, there exists a point m E n la, and since u Fa is dense in L00(G ), m is a G-invariant mean on Lco(G). By Proposition 7.2. 3(b), there is a net
J (n(g)q;J -
each L 1 (G) has the norm topology. Then the weak topology on E is the product
1 36
Ergodic theory and semisimple groups
topology for the weak topology on L 1 (G) Define a linear map T: U(G) -> E by (Tcp)9 = n(g)cp - cp. The assertion of the preceding paragraph, that (n(g)cp J - cp J) -> 0 weakly for all g, implies that 0 is in the weak closure of T(P(G)). Since T(P( G)) is clearly convex, it follows that 0 is in the strong closure of T(P(G)), or in other words, there is a net t/1 jE P(G) such that 1 1 n(g)t/JJ - t/1 Jl l 1 -> 0 for each g E G . (This argument is due to [Namioka 1] ) Finally, we wish t o pass from L 1 t o L 2 Suppose that for g E G , and B > 0, there is t/J E P( G ) c L 1 (G) such that 1 1 n(g)t/1 - t/J II 1 < B Let cp = t/1 1 12 . Then cp is a unit vector in U(G), and J i cp(xg) - cp(xW ;;:; fl t/J(xh) - t/J(x)l .. (This last asser tion just follows from the fact that for a, b � 0, la - b l 2 ;;:; la2 - b21 ) Thus 1 1 n(g) cp - cp l l z ;;:; £ 1 12 From this and the preceding paragraph it follows that the regular representation almost has invariant vectors . This completes the proof of 7 . L8 for G discrete We now turn to the proof of 7.. 1 8 for general locally compact (separable) G. If we examine the proof for G discrete, we see two problems in extending it to the continuous case. The first problem is that it is not immediately clear that there is a G-invariant mean on L 00(G), because the action of G on L 00(G) is not continuous (in G), and hence the argument in the discrete case will not apply directly. The second problem is that to establish that the regular representation almost has invariant vectors, we need to control 1 n(g)cp - cp ll z uniformly for g in a compact set In the discrete case, it therefore sufficed to do this pointwise in G.. Thus, this part of the argument also requires modification. It will be convenient now to switch to the left action of G on itself. Thus, regular representation will now mean left regular representation (i.e., (n(g)f)(x) = f(g - 1 x)) with respect to left Haar measure, and similarly the action on spaces of means will be derived from the left translation on spaces of functions. We recall that f: G -> C is called (left) uniformly continuous if for all B > 0, there is a neighborhood U of e E G such that lf(yx) - f(x) l < �> for all x E G, y E U. Let UCB(G) denote the space of (left) uniformly continuous bounded functions on G.. Then we have UCB(G) c L00(G ), and the action of G on UCB(G ) is continuous . UCB(G) is not separable, but arguing as in the first paragraph of the proof of 7. 1 . 8 for G discrete we see, using separability of G, the following: 7.2.5
Lemma: If G is amenable, then there is a G.-invariant mean on UCB(G).
We now wish to pass from a mean on UCB(G) to one on L00(G). We shall do this by smoothing an L 00-function to obtain a uniformly continuous one. If cp, f are measurable functions on G, we recall that cp *f is defined to be the function (cp *g )(x) = J cp(g)f(g - 1 x)dg, as long as this exists for almost all x E G.. We recall some basic facts concerning convolution.
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Kazhdan's property (I)
7.2.6 Proposition: (a) IffE L00(G), cp E P(G), then CfJ *fE UCB(G) (b) Ifcpn , cp E L 1 (G) and (/Jn ---> cp in norm, then CfJn * f ---> cp *f in L 00 (G)for alljE L00(G). (c) For any G, there exists an approximate identity in L 1 (G), i..e. a sequence en E P(G) such that for all cp E L 1 (G), en * cp ...... cp and cp * en ...... cp in L 1 (d) For cp E e (G), fE L00(G), and g E G, CfJ * (n(g)f) = [A(g- 1 )(p(g- 1 )cp)] *f, where A is the modular function of G, p is the representation of G on functions given by (p(g)cp)(y) = cp(yg) (and n, is as above, the left regular representation). (e) (n(g)cp) * f = n(g)(cp * f).
If G is discrete, cp E P(G) and f E L "'(G), then cp * f =
cp(g) � 0, 'I cp(g) geG
=
'I cp(g)(n(g)f).
gEG
Since
1 , it follows that for an invariant mean m, we also have
m((p * f) = m(f). We shall need this property for arbitrary G, for the mean in Lemma 7 . 2 5 .
.
If m is a G-invariant mean on UCB(G), then for cp E P(G), /E UCB(G), we have m(cp * f) = m(f). 7.2.7
Lemma:
Proof: [Greenleaf 1 ] The result will follow horn our discussion of the barycenter construction following the statement of Proposition 4.. 1 A and a formal interpre
tation of the averaging effect of convolution as a barycenter construction. Namely, we first remark that it suffices to prove the result if supp(cp) is compact Fix such a cp and fix /E UCB(G). Define a map F : G ---> UCB(G) by F(g) n(g)f Since f is uniformly continuous, F is continuous.. Let 11 be the Haar measure on G and dv cpdJ1, so that v is a probability measure on G with compact support. Let K = supp(v)( = supp(cp)) . Then F(K) c UCB(G) is norm compact, and hence so is its convex hulL The measure F.(v) is thus a probability measure supported on a (separable) compact convex set in UCB(G).. Thus, we can let h b(F.(v)) be the barycenter, h E UCB(G). We recall that h is characterized by the condition that for all A E UCB(G) * , 2(h) = JA(y)d(F.v)(y).. We claim that in fact h = cp *f· To see this, for x E G, let Ax E UCB(G)* be evaluation at x Then =
=
=
.
..
h(x) = )ox(h)
=
J2x(y)d(F * v)(y)
J2x(F(g))dv(g) = Jf(g- 1 x)cp(g)dJ1(g) =
=
(cp * f)(x).
Ergodic theory and semisimple groups
1.38
To see that m(cp *f) = m(f), we can now simply observe that since
m E UCB(G) * ,
m(h) = Jm(y)d(F* v )(y) = Jm(F(g))dv(g) = Jm(f)dv(g)
by G-invariance of m
= m(f) We can now obtain an invariant mean on L 00 (G ).
then there exists a G-invariant mean m on L '"'(G) which satisfies the further condition that m(cp * f) = m(f)for all cp E P(G), fE L 00(G) 7.2.8
Lemma: If G is amenable,
Proof: Let m be an invariant mean on
UCB(G) (Lemma 7..2..5) . Fix any ljJ E P(G) and for f E L 00(G), define m( f) = m(ljf *f) . (This is defined by 7. 2. 6a. ) Since m is a mean, so is m. We claim that if cp E P( G), m(cp * f) = m(f).. Namely, if {en} is an approximate identity (7..2..6c), m(cp * /) = lim
m(cp Hn * /)
= lim m(ljf * cp Hn * f) =
lim m(en *f) by Lemma7.2 .7 and the fact that en *fE UCB(G) by 7. 2. 6a.
= lim m(ljf * en * f) for the same reason = m(ljf *f) = m(f). To see G-invariance of m, let g E G, fE L00(G). Fix cp E P(G). . Then m(n(g)f) = m(cp * n(g)f) by the preceding paragraph. By 7.2..6d, cp * n(g)f = rx *f where rx(x) = Ll(x) - 1 cp(xg - 1 ).. However, rx E P(G), and hence, m(n(g)f) = m(rx *f) = m(f) by the preceding paragraph, completing the proof With these preparations, we now proceed to the proof of Theorem 7.. 18 in general. Proof of Theorem 7.1.8: In light of Theorem 7.2.4, it suffices to show that if G is
amenable, then the regular representation almost has invariant vectors. Choose a G-invariant mean m as in Lemma 7..2. 8. By Proposition 7.2 ..3(6), there is a net CfJ i E P(G) such that m
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Kazhdan's property (T)
/E L w(G) implies that for each cp E P(G), (q H cp1 - cp1) � 0 weakly in L 1 (G). The reason we use this latter condition rather than only use G-invariance is to be able to deal with the second problem we described above concerning generalizing 7. 1 . 8 from discrete groups to general groups, namely the problem of obtaining control uniformly on compact sets in G rather than the pointwise control which sufficed in the discrete case. As in the discrete case, we now pass to strong convergence . Let E
=
f1 L 1 (G)
cpEP(G )
and define T: P(G) � E, by ( Tcp)tp = ljJ * cp - cp. The fact that ljJ * cpi - cp1 --. 0 weakly for all ljJ E P(G) implies that 0 is in the weak closure of T(P(G)). Since T(P(G)) is convex, it follows (as in the discrete case) that 0 is in the strong closure of T(P(G)), i . e. there is a net 1/J i E P(G) such that 1 1 !/J * 1/Ji - 1/J i ll 1 � 0 for all !/J E P(G). Now let e > 0, K c G compact We claim there is an (e, K)-invariant vector in P(G) c L 1 (G), i e , cp E P(G) such that 11 n(g)cp - cp 11 1 < e for all g E K. I n fact, choose any f3 E P( G).. We claim that for some j, cp = f3 * ljJ i satisfies this condition . There exists an open neighborhood E of e E G such that 11 rr(g)/3 /3 1 1 1 < e for all g E E. Let cpE E P(G) be the normalized characteristic function of E. Then it follows that 1 1 lpE * f3 - fJ II 1 < e as welL Choose x 1 , . , Xn E G such that K c u x;E Then .
.
1 1 n(g)(/3 * 1/J i) - f3 * 1/Ji ll 1 � 1 n(g)( /3 * 1/Ji) - lpx,E * f3 * !/JJ II 1
+ 1 1 lpx,E * fJ * 1/J j
1/J j ll 1 + 11 !/J j - /3 * 1/J i ll 1 Since 1 1 !/J * ljJ i - 1/J i ll 1 --> 0 for all ljJ E P(G), for j sufficiently large the last two terms will be at most e, independent of i. Thus, (using 7.2..6e), -
Furthermore, 1 1 rr(g) /3 - lpx,E * fJ II 1 � 1 1 rr( g) /3 - rr(x;) /3 1 1 1 + 11 rr(x;) /3 - lpx,E * fJ II 1 � 1 1 rr(x ;- 1 g) /3 - fJ II 1 � 1 rr(x;- 1 g)[J - fJ II 1
+ 1 f3 - n(x;- 1 )cpx,E * fJ II 1
+ 1 1 f3 - lpE * fJ II 1
Choosing x; such that g E x;E, each of these terms is at most e, and smce 1 1 !/Ji ll 1 = 1 , we deduce that 1 rr(g)( /3 * 1/1 i) - ( /3 * 1/Ji) ll 1 � 4e Finally, we can pass from L 1 to L 2 exactly as in the discrete case and this completes the proof of Theorem 7. 1 .8 . 7.3
Unitary representations and semi-direct products,
11
In proving the vanishing of matrix coefficients for unitary representations of simple Lie groups in Chapter 2, a basic role was played by an analysis of some
140
E rgodic theory and semisimple groups
of the representation theory for certain subgroups which were actually semi direct products. For the proof of Kazhdan's result that simple Lie groups of higher IR-rank have the Kazhdan property (7J A), we will need further information on representations of semi-direct products, and we develop these results in this section We recall the context of section 2. 3. Let G be locally compact and A c G a normal abelian subgroup with A � IR" . We then have the following basic connection between representations of G and the action of G on A � lP" . If H c G is a closed subgroup, and r5 is a unitary representation of H, we denote the representation of G induced from r5 (section 4..2) by ind*(r5). .
7.3.1 Theorem [Mackey 3]: Suppose the G-action on lR" is smooth.. Then for any irreducible unitary representation n of G, there is a point Ao E lR" with G;.0 its stabilizer in G, and an irreducible unitary representation r5 of G ;.0 such that
(i) n = ind8�.0(r5), (ii) rJ I IR" = (dim r5)Jc o (iii) nllRn � n(ll,Yf;.) (section 2.3) where J1 is a measure on lR" corresponding to a quasi-invariant measure on the orbit GjG;.0. Proof: We write n I IRn � n(11,£;.l as in Proposition 2.3. 3, and apply Proposition
2.3 . .5. Since n is irreducible, 2.3..5 implies that J1 is quasi-invariant and ergodic under the G-action and dim :Yt;. is essentially constant Therefore, we can assume :Yt;. = :Yt0 a.e. for some fixed Hilbert space :Yto By smoothness of G on lR", J1 is supported on a G-orbit, and hence we can identify L 2( 1Rn, Jl, { Jlt';.}) with L2( G/G ;.0, Yl' 0) for any Jc0 E lR" in the G-orbit supporting J1 Let w be the representation of G on L2(G/G;.0, Yl'0) defined by inducing the identity representation of G;.0 on Yl'0, i . e., (w(g)f)(Jc) = p(Jc, g) 11 2f(Jcg) where p : G/G;.0 x G -> IR + is the Radon-Nikodym cocycle. (See the discussion pre ceding Proposition 4..2.20.) IfE c G/G;.0, and :YtE c L2(G/G;.0, :Yt0) is the subspace of functions supported on E, it follows that w(g)Yl'E g = Yl' E· Since we also have n(g):YtE 9 = Yl'E by Proposition 2.3.5 (after switching to a right action in 2.3.5), it follows that for all E c G/G;.0, n(g)w(g) - l Yl'E = ;YtE· However, if (M, Jl) is a (standard) measure space, and T is a unitary operator on L2(M, Jl, Yl'0) such that TYl'E = Yl'E for all E c M, then there is a Borel function M ·-> U( :Yto) m -> Um E U(Yl'o ), such that ( Tf)(m) = Um(f(m)). (See [Dixmier 2], for example.) Thus, for each g, we have [(n(g)w(g)- 1 )f](Jc) = a(Jc, g)f(Jc) for some a : G/G;.0 x G -> U(Yl'o ), and replacingf by w(g) f, we obtain (n(g)f)(Jc) = a(Jc, g)p(Jc, g) 11 2j(Jcg). Since n is a representation, it follows (as in section 4.2) that a is a cocycle
141
Kazhdan's property (T)
By Proposition 4..2J 5 there is a strict cocycle a' such that for all g, a(A, g) = a'{}o, g) a.e. It follows that a' corresponds to a representation 0' : G ;,0 --> U(Jfl' 0 ) by Proposition 4.2.1 3, and by the definition of induced representation, n = ind8.'·0(0'). Since inducing preserves direct sums and n is irreducible, (J must be as well To see the second assertion of the theorem, we observe that if g E IR", g acts trivially on lR", and hence on G/G i.o · Thus, for t E IR" we have (n(t )f)(A) = a'(A, t )f(A.) a e. for any /E L2(G/G;,0, Yl'o).. Identifying U(G/G;,0, Yl'o) with U(IR", fl, £ 0 ), we also have (n(t )f)(),) = A(t)f(A) by definition of n(JL, K,) Thus, for each t E IR", we have a'(A, t ) = A.( t) for almost all A (identirying A(t)EC as a scalar operator on Jfl'0 ). By Fubini, for 11-almost all A E lR" this equality is true of almost all t E IR", and since for fixed A, a'(A, t ) and A(t ) are continuous in t, for almost all A this equality holds for all t. (a' is continuous in t because a' is a strict cocycle, t acts trivially on lR", and hence t - > a'(A, t ) is a measurable homo morphism and hence continuous (Appendix B)). Fix Ao in the orbit supporting f1 such that a'(Ao, t) = ), 0 (t ) . By definition, 0' : G .<0 --> U(Jfl' 0 ) is given by O'(g) = a'(Ao, g), and hence it follows that O' I IR" = (dim 0') Ao 7.3.2
Example: Let N be the Heisenberg group, i . e.
[
] [
]
Let A = { g E N i a = 0}, so that A � IR2. Then g E N acts on A by
1 0 z 0 1 .y 0 0 1
1 0 z- ay 1 y 0 0 1 If (a, [3) E IR2, so that A(a , pJ(y, z) = ei(ay + Pz) then identirying A with IR2 we have (g · A(a,p))(y, z) = A(a,p) (y, z - ay) = ei
g=
0
two types of orbits. Each point on the x-axis is an orbit, and each horizontal line except for the x-axis is an orbit By 21.1 2 the N-action on lR2 is smooth. If n is an irreducible representation of N with n i A = n(JL, K , ) where f1 is supported on a point on the x-axis, then n i [N, N ] is trivial, since [N, N ] = {g E N I a = b = 0}.. Thus, n factors to an irreducible representation of N/[N, N ] � IR2, and therefore n is ! -dimensionaL On the other hand if f1 is supported on a horizontal line which is not the x-axis, then the stabilizer in N of any point in this orbit is just the group A. Thus n = ind�(O') where 0' i s an
Ergodic theory and semisimple groups
142
irreducible representation of A, and hence is one-dimensional. Therefore, in conclusion, we deduce that any irreducible representation of N is either ! -dimensional or is induced from a ! -dimensional representation of A Example: Let H = SL(2, IR}I>< IR2 where the action of SL(2, IR) on IR2 is given by usual matrix multiplication. The action on � 2 is just the adjoint action. Thus there are only two orbits of H acting on � 2, namely the origin and its complement If n is an irreducible unitary representation of H such that n I IR2 � n(Jl, f{',J with f.1 supported on the origin then n i 1R2 is trivial, and hence n factors to a representation of SL(2, IR). If f.1 is supported on the complement, the stabilizer of a point in the complement can be taken to be isomorphic to the Heisenberg group N (This can easily be seen by realizing H as a subgroup of SL(3, IR).. Namely, consider 3 x 3 matrices of the form 7.3.3
[ I �] g
o
oI 1
where g E SL(2, IR).. This group is clearly isomorphic to H in such a way that SL(2, IR) c H corresponds to matrices with b = c = 0, and IR2 c H to matrices with g = I.. The stabilizer of X E � 2, x(c, b) = eib, is readily seen to be N.) Thus in this case n = ind�(o') where a is an irreducible representation of N. Therefore we deduce that if n is any irreducible unitary representation of H, either n I IR2 is trivial, or n is induced from a representation of N. 7.3.4 Example: We can generalize 73.3 as follows. Let G = SL(2, IR) or PSL(2, IR) and suppose G acts on IRn by a rational representation such that the only fixed point under G is the origin. Let H = G1>< IRn . If n is an irreducible
unitary representation ofG then either n i iRn is trivial or n � indZ0(a) where H0 c H is an amenable subgroup. To see this, using Theorem 73.1 as in Example 7.3.3, and 4 L6(b), it suffices to see that the stabilizer in G of any point in � n except the origin, is amenable. But since there are no fixed points other than the origin, any stabilizer must be (the real points of) an IR-algebraic group of dimension at most 2 . Any such connected group is amenable, and thus the stabilizer is a finite extension of an amenable group, and hence is amenable (Corollary 4. 17). ..
We remark that the same arguments yield analogous results in 7..31-7.. 3.4 if IR is replaced by any local field of characteristic 0.. To apply results on semidirect products to showing that certain groups have Kazhdan's property, it will be useful to generalize somewhat the notion of a representation almost possessing an invariant vector. Namely, we think of almost possessing an invariant vector as almost containing the ! -dimensional
Kazhdan's property (T)
143
identity representation, and we will extend this definition by replacing the ! -dimensional identity by an arbitrary unitary representation. Let G be a locally compact group and n a unitary representation on a Hilbert space :ff, Let { v; I i = 1, . , n} be an orthonormal set in :ff , Define a function /n,{ v ,) : G ---+ M(n x n, C)(the n x n complex matrices) by (/n, {v;)(g))ij
= ( n(g)v ; I VJ )
We then call a function of the form /n , {v,J an (n x n) submatrix of n. If {vi} consists of a single element, then a 1 x 1 submatrix is of course j ust a matrix coefficient in the sense of chapter 2.. For (aij) E M(n x n, C) let 1 1 (ai1) 11 = max l aii l 7.3.5 Definition: (i) Let a, n be unitary representations of G . Let e > 0 and K c G compact We say that a is (e, K )-contained in n if for every n x n submatrix f of a, there is an n x n submatrix h of n such that 1 1 f(g) - h(g) 1 1 < e for all g E K (ii) We say that a is weakly contained in n and write a -< n, if a is (e, K ) contained in n for every e, K
Since for any unit vector ll n(g)v - v ll 2 = 2 - ((n(g)v l v) + (n(g - 1 )v l v ) ), it follows that the ! -dimensional identity representation is weakly contained in n if and only if n almost has invariant vectors in the sense of 71 1 . The notion of weak containment is due to [Fell 1 , 2] As an example, we can extend Theorem 71 . 8.
Proposition: (a) Suppose G is amenable. Let n be the regular representation. Then for any representation a of G, a -< oo n. (b) Suppose G is a group with regular representation n. If I -< oo ·· n, then G is amenable. 7.3.6
I, we have I -< n.. It follows directly from the definitions that this implies that I ® a -< n ® a, ie , a -< n ® a To prove the proposition, it then suffices to observe that n ® a � (dim a)n. However, the map U : L 2 ( G) ® :ff" --+ - L2 ( G, :ff") defined by [ U(f® v)](g) = f{g)(a(g)v) is easily seen to be a unitary equivalence of n ® a and the representation ii on L 2 (G, :ffa) given by (ii(h)cp)(g) = cp(gh), and it = (dim a)n. (b) For any cp E L2 (G), define Acp(g) = (n(g) cp l cp) , so Acp E L00(G). In the proof of Proof: (a) By 71 . 8, for the ! -dimensional identity representation
Theorem 7. 2.4, we began with the assumption that there was a sequence of unit vectors lji1 E L2 (G) such that 11 n(g)ljl1 - 1/11 11 2 ---+ 0 as j ---+ oo, uniformly on compact subsets of G, or equivalently A."'ig) ---+ 1 uniformly on compact sets of G. However, if we examine the proof, we see that in fact it suffices to find unit vectors ljl1 such that .:1."'/g) ---+ 1 only as elements of (U(G)) *, i e. , in the weak- * -topology in L 00(G). For if we have weak- * convergence, we can define cp1 as in 7.2.4 and deduce
E rgodic theory and semisimple groups
144
that (g ----> 1 1 n(g - 1 )
l m(n(g)f - f) I � 1 f 1 1 lim 1 n(g - 1 )
and hence for any positive a E L 1 (G), Ja(g) I m(n(g)f - f) I = 0 Thus m(n(g)f - f ) = 0 and this shows the existence of an invariant mean. The remainder of the proof of Theorem 7. . 2 . 4 then carries through without change It therefore suffices to show that I -< eo n implies that we can find unit vectors lj11 such that ),'�'ig) ----> 1
in the weak-*-topology on L 00(G). Let A = P'�' I
i
i
uniformly on compact sets in G. In other words, as n ----> eo, L (n(g)vnd Vni ) ----> 1
i
uniformly on compact sets, or L 1 Vn; 11 2 ),w" ,(g) ----> 1 uniformly on compact sets
i
where Wni = Vni/ 1 1 Vni 1 . Since for each n, L 1 Vni 11 2 = 1 , it follows that for the function 1 E L00(G) we actually have l E co(A). However, 1 is clearly an extreme point of Lw(G)l , and hence is an extreme point of co(A). By our remarks above, this implies 1 E A, completing the proof We shall need the following two general results . .
7.3.7 Proposition: Suppose that H c G is a closed subgroup and that a,n are representations of H with a -< n . Then indZ(a) -< indZ(n).
y : G/H ----> G be a Bore! section with y(K) compact whenever K is compact (Appendix A). We have a cocycle a : G/H x G ----> H given by a(y, g) = y(y)gy(yg) - 1 . We recall that indZ(a) is defined by ((indZ(a)(g))f)(y) = p( y, g) 1 12 a(a(y, g))f(yg) where /E L2(G/H, £'11) and p is the Radon-Nikodym Proof: Let
derivative.. (See section 4..2.) IndZ(n) is of course defined similarly. For notational convenience, set indZ(n) = U ", and similarly define U" To show that any submatrix of U 11 can be approximated by submatrices of U ", it suffices by standard approximation arguments to show this is true for a submatrix of u a defined by n (n finite) orthonormal functions fi : G/H ----> Yl'a such that fi is compactly supported and f; takes on only finitely many values . Fix such f1 , . . . , In and let K c G be compact and t: > 0. .
145
Kazhdan's property ( I)
Choose Y c G/H compact such that [i = 0 on G)H - Y for all i. Then there is a compact set K' c H such that for y E YK - 1 and g E K, cx( y, g) E K '. Let M = max { 1 fi(GIH ) 1
i
spanned by
ro ,
UN GIH). i
1 } Let V c Yf be the finite-dimensional subspace a
Choose an orthonormal basis of V and let
qJ
be the
corresponding submatrix of CJ Let If; be a submatrix of rr such that 1 1 1/J (h) ({J(h) 1 < siM 2 n for all h E K '. Let W c Yf, be the space spanned by the ortho normal vectors defining If;. By mapping one orthonormal set to the other, we obtain a unitary mapping T: V--+ W An elementary calculation then shows that for any x, V E V,
2 I < CJ(h)x, v) -
for all h E K '.
Define ai : GIH --+ £', by ai( Y) = T(fi(y)). It then follows by taking h last inequality (we can clearly assume e E K') that
=
e in the
J G!H I
7.3.8
Proposition: 2.3).. If A is
(section
Suppose nx are unitary representations of G and rr = J�rrx a representation with nx -< A for almost all x, then rr -< oo X
Proof: It is straightforward that rr -< JGl Ax where then JGl Ax � n A for n = dim L 2 (X ).
Ax = X However, if )ox = A,
We can now present a result concerning unitary representations of certain semidirect products that will arise in the next section in the proof of Kazhdan's theorem
Ergodic theory and semisimple groups
146
7.3.9 Theorem: Let G = SL(2, IR) or PSL(2, IR) and suppose G acts on IR" by a rational representation so that the only G-invariant vector is the origin. Let H = Grx IR" be the associated semidirect product. Let n be a unitary representation of H so that n I IR" has no non-trivial invariant vectors Then n -< oo ) where ). is the regular representation of H .
Proof: Write n = f1l nx where nx are irreducible (2. 3J) . Since n i iR" has no non trivial invariant vectors, for almost all x neither does nx I IR" by (2.3 ..2) . By 7.3. 8, it suffices to prove the result for all nx, i e , we may assume n is irreducible. By Example 7.34, n � indZ0(o") where Ho c H is amenable. It then follows horn Proposition 7.3 . 6 that CJ -< oo ·· AH0 where 2Ho is the regular representation of H 0 We then conclude from 7. 3..7 that n -< indZ0( oo ).H0) = oo · indZJ2H0 } However, in general the induced representation of a regular representation is the regular representation, so n -< oo ·· 2. .
7.4
Kazhdan's property for semisimple groups
The main purpose of this section is to prove Kazhdan's theorem (7. 1 .4 ). We begin with the following simple remark. 7.4.1 Lemma: If 0 -+ K -+ G -+ H -+ 0 is an exact sequence of locally compact groups, and K, H both have the Kazhdan property, so does G .
Proof: Suppose n i s a unitary representation o f G which almost has invariant vectors. Then n I K has the same property and since K has Kazhdan's property, n I K has non-trivial invariant vectors. Since K is normal in G, the space of n(K) invariant vectors is a n(G)-invariant subspace. Let n = n0 EB n 1 , where no is the subrepresentation of G on the space of all n(K)-invariant vectors. Now n 1 cannot almost have invariant vectors, for otherwise n 1 1 K would have invariant vectors which contradicts the choice of no Therefore, since n almost has invariant vectors, it follows that n0 must have this property as well But no factors to a representation of H, and since H has the Kazhdan property, n0(H ) must have non-trivial invariant vectors. Thus, so does n0(G) (and hence n(G)). 7.4.2
Theorem [Kazhdan 1]: Let G be a connected semisimple Lie group withfinite
center such that every simple factor of G has IR-rank at least 2. Then G has the Kazhdan property
147
Kazhdan's property (T)
Proof: By Lemma 7.4J, it suffices to prove this if G is simple, with trivial center,
and IR-rank(G) � 2 . We can therefore assume G is one of the form G� where G is IR-algebraic. Any such group contains a subgroup H isomorphic to a semidirect product considered in Theorem 7.3 . 9. (This is clear for SL(n, IR), since SL(3, IR) contains SL(2, IR) ><1 IR 2 as described in Example 7. 3.3 . In general, a suitable subgroup of a maximal proper parabolic subgroup will suffice . ) Let rr be a unitary representation of G which almost has invariant vectors. Suppose, however, that rr has no invariant vectors. By Moore's theorem (2..2. 1 9), rr ! IR" has no invariant vectors (where we write IR" c H as in 7. 3. 9) and hence, by Theorem 7. 3 . 9, n i H -< oo AH where AH is the regular representation of H. Since n almost has invariant vectors, I -< rr where I is the ! -dimensional identity representation, and hence I -< (rr I H ). Thus, I -< oo AH. By Proposition 7.3. 6, this implies that H is amenable, which is impossible (using 41 . 6, 4. L8) since SL(2, IR) c H or PSL(2, IR) c H.. This shows that rr must have non-trivial invariant vectors, completing the proof of Theorem 74.2. To deal with lattices, we have the following.
[Kazhdan 1]: Suppose G has the Kazhdan property and that r c G is a lattice subgroup. Then r has the Kazhdan property. 7.4.3
Proposition
Proof: Suppose rr is a representation of r with rr >- I. Then by Proposition 7 .3 . 7, ind�(n) >- ind�(J) Since Gjr has finite invariant measure, the constant functions are in L 2 (G/r), and hence I ;;:; ind�(J) Thus, I -< ind�(n), and since
G has the Kazhdan property, ind�(n) has invariant vectors . It is well-known that this implies that rr has invariant vectors. (To prove the last assertion, let rr be a representation on :If" and let a : Gjr x G --+ U(:lf ) be a cocycle corre sponding to n (4..213).. If I ;;:; ind� (rr), then there is an a-invariant (L2 ) function cp : Gjr --+ :If" Since G is transitive on Gjr, we must have that on a conull set cp # 0, and since a is unitary, rjJ : Gjr --+ ((f h the unit ball in :If"' defined by r/J(x) = cp(x)/11 cp(x) I I , will also be a-invariant By 4.. 2.19 there is a n(r)-invariant vector in (:If nh . ) Theorem 74. . 2 and Proposition 7.4.3 together yield the proof of Theorem 7 . 1 A.. There are natural examples of groups which are not semisimple but which have Kazhdan' s property.. For example: n
n
7.4.4 Theorem [ Wang
,
1]: For n � 3, G = SL(n, IR) � IR" has Kazhdan's property, where SL(n, IR) acts on IR" by matrix multiplication.
148
Ergodic theory and semisimple groups
n be a representation of G on £'" with n >- I. Suppose n has no invariant vectors. Let £' 0 = { v E £'" l n(g)v = v for g E IR" } and £' 1 = £' � . Since IR" c G is a normal subgroup £' 0 and £' 1 are invariant under n( G), and we let n0 and n 1 be the corresponding subrepresentations. From the fact that n >- I, it follows that ni >- I for either i = 0 or 1 If n0 >- I, then (no I SL(n, IR)) >- I, and by Theorem 7 4. 2, there are no I SL(n, IR) invariant vectors in £' 0 , which implies n "?;_ I Suppose on the other hand, that n 1 >- I. Let n 1 = JEilnx where ny is irreducible (2.31).. Since n 1 I IR" has no invariant vectors, it follows that for almost all x, nx I IR" has no invariant vectors (2. 3..2). The action of SL(n, IR) on iR" has two orbits, namely the origin and its complement Applying Theorem 73. 1 to the representation nx, since nx I IR" has no invariant vectors (almost all x), it follows that for almost all x, the orbit that arises must be the complement of the origin. Therefore, for almost all X, nx = ind8a(ITx) where Go is the stabilizer in G of a non-0 point in !Rn However, Lebesgue measure on iR" is an invariant measure on G/G 0 , and since this measure is not finite, it follows that nx I SL(n, IR) does not have non-trivial invariant vectors . Thus n 1 I SL(n, IR) does not have non trivial invariant vectors. However, n 1 >- I, so (n1 1 SL(n, IR)) >- I, and by Theorem 7A..2, n 1 (SL(n, IR)) has non-trivial invariant vectors . This contradiction shows that n 1 >- I is impossible, and this completes the proof Proof: Let
[ Wang 1]: For n "?;_ 3, SL(n, Z ) t>< Z " has Kazhdan's property.
7.4.5
Corollary
Proof:
7 43 and 7AA.
8
8.1
Normal Subgroups of Lattices Margulis' finiteness theorem - statement and first steps of proof
The point of this chapter is to prove the following finiteness theorem of Margulis.
8.1 . 1 Theorem [Margulis 6]: Let G be a connected semisimple Lie group with finite center and no compact factors, and let r c G be an irreducible lattice Assume IR-rank(G) � 2 . Let N c r be a normal subgroup such that f'/N is ndt amenable. Then N c Z(G) (the center of G) and in particular N is .finite. .
Combined with Corollary 7. 1. 1 2 and Theorem 7. L1 3, this yields the following result
Theorem (Margulis-Kazhdan): Let G be a connected semisimple Lie group with finite center and no compact factors, r c G an irreducible lattice. Assume IR rank( G) � 2 . Let N c r be a normal subgroup. Then either 8.1 .2
(a) N c Z(G), and so N is finite; or (b) rjN is finite. Remark: We recall that while we proved Corollary 7.1. 1 2 in Chapter 7, we did not present a proof of 7.. 1 . 1 3.. Thus, after we prove 8. L 1, we shall have a complete proof of 8. . L2 when the IR-rank of every simple factor of G is at least 2 . The remaining case is complete once 7.. L 1 3 is proved, and we again refer the reader to [Margulis 7] for this proof. Earlier results in this direction were obtained by [Raghunathan 4] by different methods.. .
Margulis' proof of Theorem 8 . U is based on his discovery of a fundamental measure theoretic property of the action of r on G/P where P c G is a minimal parabolic subgroup . Namely, suppose P' => P is another parabolic subgroup. Then clearly there is a measure preserving r-map G/P -+ G/P'. Margulis dis covered that under suitable hypotheses, any r-space X for which a measure class preserving r-map G/P -+ X exists must be of this form .
1 50
Ergodic theory and semisimple groups
6]: Let G be a connected semisimple Lie group with no compact factors, trivial center, and IR-rank(G) ;;::; 2 . Let P c G be a minimal parabolic subgroup and f c G an irreducible lattice . Suppose (X, J.L) is a (standard) measurable r -space and that there is a measure class preserving f-map rp GIP --+ X (possibly defined only a . e.). Then there is a parabolic subgroup P' :=J P so that as r -spaces, GIP' and X are isomorphic in such a way that rp corresponds (a. e ) to the natural [-map GIP -+ GIP' ( We assume GIP, GIP' to have the G.-invariant measure class.) 8.1 .3
Theorem [M argulis
.
Let us see how this result implies the finiteness theorem
G by its center, we may assume that G is center free and wish to show that N is triviaL Since rIN is not amenable, there is a compact metric rIN-space X
Proof of Theorem 8.1.1 from Theorem 8.1 .3: With G as in 8. 1 . 1 , dividing
without an invariant probability measure . We may also consider X as a compact metric f-space (on which N acts trivially).. The f-action on GIP is amenable (Example 4 3. 8), and hence by Proposition 4.3 .9 there is a measurable r-map (defined a. e..) rp : GIP --+ M(X ) (where, as usual, M(X ) is the space of probability measures on X ). Let J1 be a probability measure on GIP quasi-invariant under G. Then rp * (J.L) is a measure on M(X ) quasi-invariant under r, and q; :(GIP, J.L) -+ (M(X ), rp * (J.L)) is a measure class preserving f-map . By Theorem 8.1 . 3, there is a parabolic subgroup P' :=J P such that (M(X ), rp * (J.L)) � GIP' as f-spaces. If P' = G, then r would have a fixed point in M(X ), which would contradict the choice of X. Therefore P' i= G. Recall that N acts trivially on X, hence on M(X ), and therefore each element of N fixes almost all points in GIP'. Since N acts continuously on GIP', this implies that N acts trivially o n GIP', so that N c Let G Then H
n gP'g - 1 , and H = n gP'g - 1 is a proper normal subgroup.
geG
geG
=
Il G i be the decomposition of G into a product of simple groups..
=
Il Gi, where J c I is a proper subset. Since N H, N is normalized
iel
c
i eJ
by Il Gi· · But since N is normalized by Il Gi and by r, it is normalized 1-J
I-J
b y the product of these groups which i s dense i n G b y irreducibility . Thus, N c G is normal, and since N is discrete and G has trivial center, N is triviaL This completes the proof. Before turning to the proof of Theorem 8. 1 3, it will be convenient to reformulate it slightly. We recall that if (X, J.L) is a measure space, by its measure algebra (or its Boolean a-algebra), B(X ), we mean the space of measurable subsets of X, two sets being identified if they differ by a null set. We can identify
Normal subgroups of lattices
!51
B(X ) as a subset o f the unit ball i n L "'(X ), namely B(X ) = { f E L aocx ) I f 2 = f(a e. )}. Then B(X ) is a weak-*-closed subset of L ao(X ). The induced topology on B(X ) is independent of the particular measure on X within its measure class. If
[Margulis 6]: Let G, P, r as in 8. LJ. Suppose B e B(G/P) is a closed Boolean subspace (i. e . , closed in topology, and closed under the Boolean operations.) If B is r-invariant, then B is G-invariant.
8.1.4
Theorem
Let us see how to deduce 8 . 13 from 8. 1 .4. Let B =
Ergodic theory and semisimple groups
1 52
Our task is therefore to prove Theorem 8. 1 .4, and we will do this in the following sections.
8.2
Contracting automorphisms of groups
In this section we examine some properties of contracting automorphisms that we will need in the proof of 8. 1 .4 8.2.1
Definition: Let H be a locally compact, compactly generated (separable) group . An automorphism A : H --+ H is called contracting if for any compact
neighborhood U of e E H and any compact subset K c H, there is an integer N such that for n � N, A n(K ) c U. 8.2.2
Example: (a) Let 0 < rx < 1 and define A : IRn --+ IR n by Ax
=
rxx.
(b) Let g be an r x r diagonal matrix with diagonal entries g;; = A;, where A; < A; + 1 . Let U be the unipotent group of upper triangular matrices with all diagonal entries equal to 1 , and let a be the transpose of U, i.e., the correspond ing lower triangular matrices. Let A : G L(n, IR) --+ GL(n, IR) be A = Int(g), i e. , A( h) = ghg - l Then both U and a are normalized by g, so that A defines an automorphism of both U and a. If h = (h;i), then (Ah);i = A;Aj 1 h;h and hence A is a contracting automorphism of U. Similarly, (A - 1 h);i = A;- 1 Ajhih and so A - 1 is. a contracting automorphism of a. 8.2.3
Example: The preceding example can be generalized as follows. Fix an integer r � 2, and positive integers r 1 < r2 < · ·· ·· < rn = r. Choose real numbers
< An and let s be the diagonal r x r matrix given by S;; = A; for = 0) . Let V be the subgroup of the upper triangular unipotent group U (of 8..2..2(b)) given by V = {u E U i u;i = 0 if i =F j and if there exists m such that r 1 < i,j � r } Then Int(s) (Le. , h --+ shs - 1 ) is a contracting automorphism of V, and Int(s) - 1 is a contracting automorphism of V, the
A1
< A2
< ···
r1 - 1 < i � r; (where we take r0 m -
transpose of
m
.
V
Using the structure theory of semisimple groups, we can find an analogue of Example 8.2. 3 in any connected semisimple Lie group with finite center and no compact factors. To express this, let us rephrase some of the struc ture in the above example. Let P0 be the stabilizer in SL(r, IR) of the flag 0 c W1 c W2 c · · · c Wn = R' where Wk = [e 1 , . . , e, J , the linear span of the
1 53
Normal subgroups of lattices
first rk standard basis vectors. Then Po is a parabolic subgroup of SL(r, IR), and V is the unipotent radical of P0 If we let R = { g E Po l g Yk = Yk for all k where Yk = [e,k _ , + 1 , , e,J }, then Po is the semidirect product P 0 = R t>< V and R is reductive (so R is semisimple after dividing by its center). We also remark that for s the diagonal matrix above, s E Z(R), the center of R. The transposed subgroup J50 is an opposite . parabolic in the sense that P o n P0 = R. The subgroup V is the unipotent radical of Po We have thus seen the validity of the following assertion for G = SL(r, IR), and which is true in general from structure theory.
Let G be a connected semisimple Lie group with finite center and no compactfactors. Let A be a maximal connected IR-split abelian subgroup of G and P a proper parabolic subgroup of G with P ::::J A.. Let V c P be the unipotent radical of P, P = R t>< V where R is the reductive component of P containing A, and V the unipotent radical of the opposite parabolic P . Then there exists s E A, with s =!= e and s E Z(R) such that the inner automorphism Int(s) is contracting on V and Int(s) - 1 is contracting on V
8.2.4
Proposition:
.
We now return to the general situation and make the following simple observation Proposition: Suppose that A is an automorphism of H and that A - 1 is contracting. Let E c H be a measurable subset. Then (a) if E contains a neighborhood of e E H, then as n ----> oo, A"(E) ---> H in measure, (b) if E misses a neighborhood of e E H, then as n ----> oo, A"(E) ----> 0 in measure 8.2.5
Proof: We recall (see the beginning of section 1.3) that En ----> E in measure means that on any subset F of finite measure we have p((En n F ) I1 (E n F)) ----> 0. Consider assertion (a). If F is any compact set, since A - 1 is a contraction, for n sufficiently large A - n(F) c E, so F c A"(E).. Since any set of finite measure differs from a compact set by a set of small measure, it follows that A"(E ) ----> H in measure Similarly, to see (b), we note that if F is any compact set, and E misses a neighborhood of e, then for n sufficiently large A"(E ) n F = 0
The main result we will need about contracting automorphisms is a general ization of 8..2. 5 to arbitrary measurable sets, without the assumption that the set either contains or misses a neighborhood of the identity.. This will follow by a suitable approximation of an arbitrary set by open sets. In IR", a useful approximation of this kind is the classical Lebesgue theorem on density points.
Ergodic theory and semisimple groups
1 54
We recall that if E c IR" is a measurable set and x E E, then x is called a point
. . Jl(B(x, r) n E) = 1 where f1 is Lebesgue measure and B(x, r) f1(B(X, r)) r�o is the ball of radius r about x. Then Lebesgue' s theorem asserts that for any .
of density for E If bm
measurable E, almost every point of E is a point of density An examination of the proof of this theorem (see [Stein 1], for example), shows that it is true in the following more general context. (The sets Bx.n below play the role of B( x , r ") in IRm )
8.2.6 Theorem (Lebesgue): Suppose X is a locally compact separable metrizable space, and that f1 is a (J-finite measure on X which is positive on open sets and finite on compact sets . Suppose that for each x E X, we have a decreasing sequence, Bx.n, of relatively compact open neighborhoods of x, forming a basis for the open sets at x, such that
(i) Jl(Bx ,n - d/Jl(Bx.n) is constant independent of x, n. (ii) If n ;;:;; p and Bx.n n By.p f= 0, then By,p c Bx,n - 2 If E c
X is any measurable set, then for almost all x E E,
lim Jl(Bx,n !I E)/Jl(Bx,n) = L
n � oo
Of course, if (X, Jl) is IRm with Lebesgue measure, as we remarked above Bx , n = B(x, r ") satisfy these conditions. More generally, the following result shows that we can obtain such sets from more general contracting auto morphisms.
Let A be a contracting automorphism of H, and suppose that U c H is a relatively compact open neighborhood of the identity with U = U - l and such that A(U 2 ) c U.. (By U 2 we mean U U.. ) For X E H, let Bx,n = xA"(U). Then these sets satisfy the hypotheses of Theorem 8. . 2 .6, where f1 is left Haar measure on H. 8.2.7
Proposition:
Proof: To see that they satisfy condition (i) of 8.2. 6, recall first that if A is an automorphism of H and f1 is Haar measure on H, then A*(Jl) is also a Haar measure and hence for some constant c, A *(Jl) = Cfl Then (i) follows from the fact that
Normal subgroups of lattices
155
To verify (ii), suppose that there exists z E yA P( U ) n xA"(U ), with n � p Let w E yA P( U ) Then ..
i e , w E zA"- 1 ( U ).. Since z E xA"( U ), it follows that
W E xA"( U )A"- 1 ( U ) c xA" - 1 ( U )A"- 1 ( U ) c xA" - 1 ( U 2 ) c xA" - 2 (A(U 2 )) c
xA" - 2 ( U ).
Thus, yA P( U ) c xA" - 2 ( U ), completing the proof Remarks: ( 1 ) When
H
=
IR", Ax = x/2, and U is the unit ball, we recover the
classical example. (2) Given any contracting automorphism, replacing A by AN for some N � 0, we can assume we have A(U 2 ) c U for any fixed relatively compact symmetric neighborhood of the origin. (3) If A(U 2 ) c U, then A((U "f) c U " for any n. Finally, we return to the promised generalization of Proposition 8.2 5 ..
.
8.2.8 Proposition [Margulis 6]: Let A be an automorphism of the compactly generated locally compact (separable) group H.. Suppose A - l is contracting. Then, replacing A by some fixed positive power of A if necessary, we have the following property. If E c H is any measurable set, then for almost all h E H,
(i) A"(hE) converges in measure to H if e E hE. (ii) A"(hE) converges in measure to 0 if e rf.: hE (2) above, we can assume A - 1 ( U 2 ) c U where U c H is a relatively compact symmetric open neighborhood of e E H, such that U U " = H. To see (i) it suffices to see that for any measurable set E c H and Proof: By remark
n� l
any open set V c H, of finite measure, we have
Ergodic theory and semisimple groups
! 56
(*)
lim Jl(A"(hE) n V)
=
11( V)
for almost all h E E - 1 .
However, if we apply ( * ) with E replaced by H - E, we see that ( * ) implies (ii) as well. To verify ( * ), it suffices to verify it for arbitrary measurable E and V = U ' for r � 1 . Thus, we wish t o show that for almost all h E E - 1 lim Jl(A"(hE) n U ' )/Jl(U ' ) = L As we observed in the proof of 8.2. 7, A./1
= CJl,
so this is equivalent to
n� eo
i e , for almost all h E E - 1 , (i. e . , a . e h- 1 E E),
However, since A - 1 is contracting, the validity of this equation follows from 8..2.. 6, 8.2..7, and remark (3) above. This proves 8.2. 8 In our application of 8.2. 8, the group H will actually be appearing as a factor in a semidirect product. Therefore, it will be useful to formulate a consequence of 8. . 2 .. 8 in that context Thus, suppose G = H ><J L, and that A is an automorphism of G which restricts to automorphisms of both H and L We shall assume that A - t is contracting on H and that A is the identity on L. Let E c G be a measurable subset We want to identify lim A"(E) in the spirit of 8.2.8. For each y E L, let Ey = {x E H l xy E E}. By 8.2 . 8, for each y, we have that for a.e . h E H, A"(hEy) --+ H (or 0) in measure on H if e E hEy (or e rj; hEy). We observe that e E hEy if and only if y E hE By Fubini, we have for almost all h E H, that for almost all y, A"(hEy) -+ H (or 0) in measure on H if y E hE (or y rf; hE). Once again using Fubini's theorem (by integrating over y) we deduce that for almost all h E H,
A"(hE) --+ H · (hE n L) in measure on G.. Suppose we have h E H for which this is true, and let y E L be arbitrary. Then yA"(hE) --+ yH (hE n L), and since A is trivial on L and y normalizes H, we deduce the following.
1 57
Normal subgroups of lattices
Corollary: Let G = H ><J L and A be as above. Then (possibly after replacing A by some positive power of A),for any E c G, we have that for almost all g E G, A n(gE) --> lj;(gE) in measure on G, where lj;(F) = H (F n L) for F c G. 8.2.9
8.3
Completion of the proof - equivariant measurable quotients of flag varieties
In this section we present Margulis' proof of Theorem 81 . 4.. As with the proof of the superrigidity theorems in Chapter 5, we shall use the fact that up to null sets, we can identify G/P with a unipotent subgroup of G (Lemma 5 1 .4).. While this was useful in chapter 5 for showing that a suitable measurable map on G/P was rational, here it will be useful as we will be able to apply the results of section 8 .2 on contracting automorphisms. We shall begin by reviewing some structural facts about semisimple groups (with the hypotheses in 81 .4), presenting SL(n, IR) as an example. In this section we shall be taking G/H to be {gH } , so that G acts on the left on G/H Let A c G be a maximal connected IR-split abelian subgroup (so for SL(n, IR) we can take A to be the positive diagonals..) Let P be a minimal parabolic subgroup containing A, and V the unipotent radical of A.. (Thus for SL(n, IR), we can take P to be the upper triangular matrices and V the subgroup of P consisting of all matrices in P with all diagonal entries equal to 1 .) Let P be the opposite parabolic, with unipotent radical V, (for SL(n, IR), the corresponding lower triangular matrices. ) Then (Lemma 5 . 1 .4), the restriction ofthe natural map G --> G/P to V defines a rational isomorphism of V with an open conull subset of .
G/P. Now let P0 ::::;, P be another parabolic. (For SL(n, IR), take P 0 to be the stabilizer of a flag as in the discussion preceding 8..24. ) Let V0 be the unipotent radical of P 0 , and P0 , V0 the corresponding opposite subgroups. One shows as in Lemma 5.1 4 that the natural map V0 --> G/Po is also a rational isomorphism onto an open conull subset of G/P0 .. In 8..14, we are interested in the map G/P --> G/P0 , and hence we wish to express this in terms of V, V0 In other words, we want to identify the map V--> V0 making the following diagram commute.
(* )
V
G/P
l(t Vo
G/Po
I I I I I I I
Let R 0 be the reductive component of
1
P
0 containing A, so P 0
=
R0 t>< V0,
1 58
Ergodic theory and semisimple groups
P0 = Ro 1>< V0 (For SL(n, IR), this group is also described in the discussion preceding 8. 2.4.) Let Lo = R 0 n V( = Po n V) (For Po c SL(n, IR), as above, Lo = { g E R 0 I g;; = 1 and g;i = 0 if i < j }.) Then V0 , L0 c V, and in fact V = V0 ><J L0 Thus, the map V ---> V0 we needed above is simply the projection of V onto V0 given by this semidirect product decomposition. Let i : V ---> G/P be the natural map, which is a measure space isomorp hism For g E G, the action of g on G/P will then give us an a.e . defined measure space automorphism of V More precisely, if g E G, then for almost all x E V, gi(x) E i( V). Therefore, we define g o x E V via the equation g i(x) = i(g x), so that for each g this is defined a.e. For certain elements of G, we can be more explicit We observe that a
(i) If g E V, then g o x = g x for all x E V. (ii) If g E P n P (i. e. , g E R where P = R 1>< V), then g o x To see (ii), note that i(gxg - 1 ) = gxg - 1 P = =
=
gxg - 1 .
gxP gi(x).
By the equation defining g o x, we obtain (ii). We shall identify the G-action on B(G/P) with this induced action on B( V ). Now let B c B( V) = B(G/P) be a r-invariant closed Boolean s ubspace, and suppose E c V with E E B Let g E G. Of course, if this implied g · E E B, we would be done. Our first reduction of the proof is to show that it suffices to show that g ·· E E B where E is a certain subset related to E. M ore precisely, suppose E c V, and let P 0 -:::J P be as above, so that V = Vo ><J Lo Define t/lo(E) = Vo (E n Lo) (Cf 8. 2.9.).
To prove Theorem 8. 1 .4, it suffices to prove that for any V E E B c B( ), g E G and parabolic P 0 , G -:::J P 0 -:::J P, we have g · t/J0(vE)EBfor almost
8.3.1
Lemma:
-
all v E V
*
*
Proof: We continue to identify B( V ) � B(G/P). If Po -:::J P we shall also identify
B(G/Po) as a subspace of B(G/P) via the natural map GjP ---> G/P0 . In light of our discussion above, this identification is identified with the realization of B{ V0 ) as a subspace of B( V ) induced by the pr�jection V --+ V0 • Let B' be the largest G-invariant closed Boolean subspace of B, so we have B' c B c B(G/P). By our remarks in section 8J B' can be identified with B(G/P')
159
Normal subgroups of lattices
where P' ::::J P. We may have P' = G (so that B' is trivial), but we can assume P' #- P for otherwise we are clearly done. We claim that B' = B Suppose not, and choose E c G/P with E E B but E rj; B(G/P') . , Pm be the standard parabolics with P' ::::J Pi ::::J P, and Pi minimal Let P1 *
with respect to this property. As above, write Pi = R i 1>< Vi Li = R i n V The structure theory of semisimple group implies that P' is generated by {Pi} (This is a straightforward exercise for G = SL(n, IR) ) We have the map G ----> GjP, and we can thus identify B(G/P) as a subspace of B(G), and since E E B(G/P), we have, as an element of B(G), that E g = E for all g E P On the other hand, since E rf; B(G/P'), we cannot have E g = E for all g E P' Thus, for some i, there is an element g E P; such that E g #- E (modulo null sets). In other words, E E B(G/P) is not in the image of B(GjPi) under the embedding B(G/Pi) -> B(G/P) defined by the natural projection . In terms of the diagram ( * ) above, with P 0 = Pi, this means that considering E E B( V), we have E [; #- E. We now consider E E B{ V), and recall that V = Vi ><J [; Since E Li #- E, it follows from F ubini's theorem that for a set of w E vi of positive measure, { y E r; � wy E E} is neither null nor conull in L;, i e ' [; n w- 1 E is neither null nor conull in L;. For such a W E vi, we clearly have for any V E WLi that L; n v - 1 E is neither null nor conull in L; . Hence, we deduce that for a set of v E V of positive measure, 1/J i(vE) E B( V ) is not right I;-invariant, hence not ?'-invariant. In other words, under the identification in diagram ( * ), 1/J;(vE) is not in the image of B(G/P;) c B(G/P), and in particular is not in the image of B(G/P') (which is B '). However, IR-rank(G) � 2, and hence P; #- G . Therefore, by the hypotheses of the lemma, we can find v E V with 1/J;(vE) rj; B', but g 1/J ;( vE) E B for all g E G. Let B be the closed Boolean subspace generated by { gi/J;(vE) I g E G} and B' . Then B c B and B is G.-invariant However, 1/J;(vE) E B - B', so this contradicts the maximality assumption on B', proving Lemma 8.JJ . ,
The completion of the proof of Theorem 8 . 1 .4, i.e. , verification of the conditions in Lemma 8.3J, will follow from the following two lemmas . 8.3.2
Lemma:
Let P0 be a parabolic with G ::::J P0 ::::J P. Write Po = R o ><J Vo as *
above, and as in 8..2.4 , choose non-trivial s E Z(R0), the center of R 0 , (with s E A as in 8.2.4) such that Int(s) is contracting on V0 and Int(s) - ! is contracting on V0 Then for any measurable set E c V, we have (possibly replacing s by sN for some N � 1), for almost all v E V, as n ----> oo, Int(s)"(vE) ----> 1/!o(vE) in measure on V Proof: Apply Corollary 8..2.9.
The second lemma we will need is an ergodicity type of statement.
Ergodic theory and semisimple groups
160
8.3.3
Lemma: Let s be as in 8 ..3.. 2.
is dense in G
Thenfor almost all V E V, { yv- 1 s - n I y E r, n ;?; 1 }
This lemma is close to being a consequence of Moore's theorem (2 ..2. 6). Namely, the latter implies that the integer action defined by powers of s acts ergodically on G;r, and hence (by 2. 1 . 7), for almost all g E G, { ygs - n I y E r, n E Z } is dense in G. Thus 8. 3 3 differs from this last assertion in two respects. First, in 8.. 33 we consider only positive powers of s - 1 , and second we have an assertion about almost all v E V The latter is of course a lower dimensional space than G, and therefore such an assertion is not an immediate consequence of M oore's theorem. We shall show how to obtain 8 ..33 from Moore's theorem, but first show why 8 ..3.. 2 and 8.1.3 suffice to prove Theorem 8. 14. Proof of Theorem 8.1.4: We verify the condition of Lemma 8.J . L Let
E c V with
E E B.. Then for almost all v, the conclusions of 83.2 and 8..33 are valid . Fix g E G. Then by 8.3 . 3, we can write g = lim Yiv 1 s - n ; where ni ;?; 1. Furthermore, j
-
since the conclusion of 8. 1 3 is valid if we replace s by some positive power of
s, we can assume ni � oo as J � oo. Let gi = Yiv - 1 s - n;, so that Yi = gisn iv. Recalling that we are denoting the G action on V and B( V) by we have Yi o E = (gisn ;v) o E = gi o (sn ivEs- n ;) by (i), (ii) in the discussion preceding 8.3.. L As j � oo, gi � g and sn ivEs - n ; � t/J 0 (vE) i n measure o n V W e thus have Yi o E � g o t/J 0 (vE) in measure, and hence in the weak- * -topology on B( V).. However, since E E B and B is closed, we deduce that g o t/J 0 (vE) E B and by Lemma o,
831 , this completes the proof of Theorem 8. 14.
It therefore remains only to prove Lemma 8 . .3 ..3 We begin with a general observation to deal with the fact that we restrict ourselves to n ;?; 1 in 8.3.3 .. 8.3.4 Proposition: Suppose the group of integers Z acts ergodically and with a finite invariant measure on a space (S, /1) Then for any Y c S with Jl( Y) > 0, U ( Y n) is conull. n� 1
W = U ( Y· n). Then W 1 c W Since the action is measure preserving n� 1 and the measure is finite, W 1 = W. Thus W is essentially invariant under the Z-action, and by ergodicity W is conull Proof: Let
Normal subgroups of lattices
161
8.3.5 Corollary: With the notation of 8.34, if S is also a separable metric space and J1 is positive on open sets, then for almost all x E S, {x ( - n) I n � 1 } is dense in S
Proof: The proof of 2. L7 applies in this case as well.
W = {g E G I {ygs -" l y E r, n � 1 } is dense in G } . By Moore's theorem (2. 2. 6), {s-", n E Z } is ergodic on Gjr, and hence by Corollary 8. 3..5, W is conulL Recall that the multiplication map V x P ---> G is an injective map onto a conull subset of G. For each V E V, let Yv = { p E P i vp E W}, and let U = { v E VI Yv is conull in P} Then by Fubini, U is conull in V To prove the lemma, it suffices to show that e E Yv for all v E U Thus, we fix v E U Then we can choose Pk E Yv such that Pk ---> e .. By the choice of W, we then have that for each k � 1 , {yvpks- " l y E r, n � 1 } is dense in G, i.e , { yvs -"s"pks -" I y E r, n � 1 } is dense in G . Recall that we can write P = R r>< V, and that R is the centralizer of A in G. (In SL(n, IR), R is just the subgroup of all diagonal matrices.) Let Pk = ukrk, rk E R, uk E V Since s E A, s commutes with rk, and hence for each k � 1 we have {yvs -"s"uks -" I y E r, n � 1 } is dense in G. Since Pk ---> e, we also have uk ---> e. Suppose e fj; Yv, so that { yvs -" I n � 1 } misses an open set A c V Then we can find an open set B c A and an open symmetric neighborhood C of the origin in V such that BC c A . Thus B n (X - A)C = 0. However, { uk} is relatively compact since uk ---> e Thus, for n sufficiently large, Int(s)"( { uk}) c C since I nt(s) is contracting on V For any n, there are only finitely many elements of Int(s)"( {uk}) which do not lie in C, and it follows that for some k (in fact for all sufficiently large k), {Int(s")uk l n � 1 } c C But since {yvs -" l n � 1 } c X - A, and (X - A)C n B = 0, it follows that for some k, {yvs - "s"uks - " 1 n � 1 } does hOt inter sect B. This is impossible because this set is dense, contradicting the assumption that e fj; Yv This completes the proof of the lemma. Proof of Lemma 8.3.3: Let
9
9.1
Further Results on Ergodic Actions Cocycles and Kazhdan's property
If a group has the Kazhdan property, then there are significant restrictions on the type of cocycles certain actions of the group can have . We begin with the following
Theorem: Let G be a group with the Kazhdan property. Suppose S is an ergodic G-space with invariant probability measure. Let a .: S x G -+ H be a cocycle where H is an amenable group.. Then a is equivalent to a cocycle into a compact subgroup . (In particular, if H = IRn X z m , a is trivial.)
9.1 . 1
For G discrete, this was first shown in [Schmidt 2], [Zimmer 7J We preface the proof with the following general observation.
9.1.2 Lemma: Let G, H be locally compact, S an ergodic G--space and a · S x G -+ H a cocycle. Let n be the regular representation of H. Then a is equivalent to a cocycle into a compact subgroup of H if and only if there is a n o a-invariant function r.p . S -+ L 2(H h , the norm one vectors in L2(H ). (Definition 4.2J 7..)
a � f3 where f3(S x G) c K, K c H a compact group, then there is a n(K)-invariant vector in L 2(H ) t (simply because K is compact), and hence a n o {3-invariant function S -+ L 2(H ) t Therefore, there is such a n a-invariant function (4.2J 8c).. To see the converse, we first observe that for each r.p E L2(H ), n(h ) r.p -+ 0 in the weak- * -topology as h -+ oo in H. (To see this, suppose !/J E L2 (H ). Then we can find compact sets A, B such that 11 XACfJ - r.p 11 and 11 XB!/1 - !/I ll are smalL But as h -+ oo , we can assume Ah n B = 0, and hence
a
163
Further results on ergodic actions
Proof of Theorem 9.1 . 1 : Let n be the regular representation of H. Then n o rx is a cocycle S x G -+ U(L2 (H )), and we can form the induced representation, say a, of G on L2 (S; L 2 (H )); namely (a(g)f)(s) = n(rx(s, g))f(sg).. We know from Theorem 71 . 8 that n almost has invariant vectors. We claim that a does as well Let K c G be a compact symmetric set and t: > 0. Let v denote Haar measure on G. If A c H is compact and g E G, let S(g, A) = { s E S i o:(s, g) E A, o:(s, g - 1 ) E A }
Since rx is measurable and H is a-compact, there is a compact set A c H such that (p,
x
v)( { s, g) E S
x K l o:(s, g) E A, rx(s, g - 1 ) E A }) ;?; (1 - e/3)(v(K)).
By Fubini's theorem, v(Ko) > 0, where Ko = { g E K i p,(S(g, A)) ;?; 1 - e/3 } .
Clearly K 0 is symmetric. Thus K 6 = K 0 K 0 1 contains a neighborhood W of the identity (Appendix B) By the cocycle identity o:(s, gh) = o:(s, g)o:(sg, h) and the fact that p, is G-invariant, for any y E K 6 we have p,(S ( y, A 2 ) ) ;?; 1 - 2£/1 Now choose g; E G, i = 1, . .. , n, such that u g; W => K. Then there is a compact set B c H such that p,(S(g;, B)) > 1 e/3 for all i . It follows as above from the cocycle identity that p,(S(g, BA 2 )) > 1 - t: for all g E K Since n almost has invariant vectors, we can choose a unit vector x E L 2 (H) such that 11 n(h)x x 1 1 < dor all h E BA 2 Define q; E L2 (S; L 2 (H )) by q;(s) = x for all s E S. Then for g E K, we have -
-
p,( { s E S I ll n(o:(s, g))q;(sg) - q;(s) 1 1 < t: }) > 1
-
£..
Therefore an elementary computation shows that 1 1 a(g)q; - q; 11 < A-(e) for all g E K where A-(e) -+ 0 as £ -+ 0.. This verifies our assertion that a almost has invariant vectors . Since G has the Kazhdan property, this implies that a has a non-trivial invariant vector. In other words, letting H act in L 2(H ) via n, there is an a-invariant function f: S -+ L 2 (H ) Finally, we remark that we can assume that for all s,f(s) E L 2 (H h by ergodicity. More precisely, since f =P O, f(s) =P 0 on a set of positive measure and since o:(s, g) f(sg) = f (s) a.e , ergodicity implies that 1 1 f(s) 11 is essentially constant The theorem now follows from Lemma 9. 1 . 2. We remark that this theorem is false if the assumption of finite invariant measure is eliminated .
E r godic theory and semisimple groups
1 64
Corollary: Let G = [1 G;, G; a connected simple non-compact Lie group withfinite center. Let S be an ergodic G-space withfinite invariant measure. Assume that G has the Kazhdan property, eg , !R-rank(G;) � 2 for all i. Suppose rx S x G -+ H is a cocycle with H amenable Then rx is equivalent to a cocycle into a compact subgroup of H (In particular, if H = !R" X zm, (X is trivial..) 9.1.3
Proof: This follows from 9. . 1 1, and 7. 1 4.
Kazhdan's property has significant implications for the kind of actions such a group can have on a compact manifold. (See [Zimmer 1 7].) The geometric background required for most of these results makes them beyond the scope of this book However, we shall be applying Theorem 9.1J in section 9.4 to study entropy of actions of semisimple Lie groups or their lattices on compact manifolds. Here, we only present a very elementary illustration of the geometric implications of the Kazhdan property, showing that groups with this property do not have volume preserving hyperbolic actions . We recall the latter notion Let M be a compact manifold and f: M -+ M a diffeomorphism.. There are a variety of conditions of f that describe "hyperbolic" behavior. Here, as an illustration we consider the following notion. See for example [Katok 1 ] for a discussion of notions of hyperbolicity. Call f weakly hyperbolic if there is a cif-invariant splitting of the tangent bundle of M into measurable subbundles, TM = E 1 + E2 such that for all v E E 1 , 1 drv 11 -+ 0 as n -+ oo and for all v E E2 , 11 dj - n v 1 -+ 0 as n -+ oo . Since M is compact, this is independent ofthe Riemannian metric If r is a group acting on M by diffeomorphisms, call the action weakly hyperbolic if the tangent bundle splits into dr-invariant subbundles as above, and with respect to this splitting, there is some y E r that is weakly hyperbolic. 9.1.4 Proposition: Suppose r is a group with the Kazhdan property Then r does not have any weakly hyperbolic volume preserving ergodic actions on a com pact manifold. Proof: Suppose we had such an action . We can measurably choose orthogonal bases for E;, and hence dj" acting on each E; will give us a cocycle rx; : S x r -+ GL(n;, !R), and for some Yo E r, and i = 1 or 2, 1 rx;(s, y�)v 11 ---. 0 for all v E !R"' . (Cf. Example 4..2.3.) Let us simply denote rx; by rx. Let P : S x r -. !R be p(s, y) = log l det rx(s, y) l . Then for all s, we have p(s, y�) -+ - oo as n -. oo. However, by Theorem 9. Ll, p is trivial, i.e., there is a function
Further results on ergodic actions
165
n � N, T"x � A } It suffices to show ,u(A N ) = 0 for all N. Suppose not Consider the sets T' N(A N) where r is a positive integer. These are all of the same measure since T preserves measure, and hence there are r1 , r2 , (say r 1 < r 2 ) such that T'' N(A N ) n T' 2N(AN) =I= 0 Therefore, there exist x, y E AN such that T'1 NX = T'2Ny, i.e., r
Proof: Let A N = {x E A i for all
As with amenability, one can extend the notion of the Kazhdan property for a group to that of the Kazhdan property for an action. Here, we shall only indicate, without proof, some developments in this direction, for simplicity of exposition restricting our attention to discrete groups. Definition [Zimmer 7]: (a) Let r be a discrete group, (S, ,u) an ergodic r-space, ,u a quasi-invariant probability measure. Let e > 0 and K c r a finite subset of r . Let yt' be a Hilbert space and IX : s X r -+ U(Yf') be a cocycle. Let F 1 (S, Yf') = {
We then have the following results. We refer the reader to [Zimmer 7] for proofs . Theorem [Zimmer 7]: (a) If r has the Kazhdan property, and S has a finite r-invariant measure, then the action ofT on S has the Kazhdan property ( This is false in general without the assumption offinite invariant measure. )
9.1.7
1 66
Ergodic theory and semisimple groups
(b) If the action of r on S has the Kazhdmi property, finite invariant measure and is weakly mixing (i . e., L2 (S) 9 C has no finite dimensional invariant subspaces), then r has the Kazhdan property.. ( This is also false in general without the assumption of.finite invariant measure. ) (c) If the action of r on S has the Kazhdan property, then any cocycle into an amenable group is equivalent to one into a compact subgroup. (Cf Theorem 9 1.1 . ) (d) For essentially free actions, the Kazhdan property is an invariant of orbit equivalence. As a consequence of 9 . L7 and 7.4..5, we have:
3, let [ 1 = SL(m, Z) x z m and [ 2 = SL(n, Z)r>
9.2
Corollary: For n �
The algebraic hull of a cocycle
In this section we introduce a useful invariant of a cocycle into an algebraic group . We shall be applying this in Section 93 to prove a basic p roperty about orbit equivalence for actions of lattices in higher rank semisimple Lie groups and in Section 9.4 to the study of entropy of the actions of these groups on compact manifolds. We will be concerned with the following question. Suppose G, H are locally compact, S an ergodic G-space and a : S x G -+ H is a cocycle. Is there a unique (up to conjugacy) smallest subgroup L of H such that a is equivalent to a cocycle taking values in L? In general the answer is no . However, if H is algebraic and we restrict attention to algebraic subgroups, then the answer is yes. 9.2.1 Proposition: Let k be a localfield of characteristic 0, H an algebraic k-group and a : S x G -+ Hk Then there is an algebraic k-group L c H such that a is equiva lent to a cocycle taking values in Lk, but is not equivalent to a cocycle taking values in any proper subgroup of Lk of theform M k, where M is also a k-group . The group Lk is unique up to conjugacy in Hk.
Proof: Such a minimal Lk exists by the descending chain condition on algebraic
groups. To see it is unique up to conjugacy, suppose we have two such minimal groups, L, J . Then by 4.2. 1 8(b), we have a-invariant functions cp : S -> Hk/Lk ,
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1/J : S ----" Hk/h Then A = (cp, 1/J ) : S ----" Hk/Lk x Hk/1k is also a-invariant However, the action of Hk on Hk/Lk x Hk/Jk is smooth (3. 1 .3).. Therefore, by the cocycle reduction lemma (52. 1 1), a is equivalent to a cocycle into the stabilizer of a point in Hk/Lk x Hk/h In any orbit, we can choose the stabilizer to be of the form Lk n hJkh- 1 where h E Hk By the minimality property of Lk, Lk c hJk h - 1 Similarly, Jk is contained in a conjugate of Lk, and this implies that Jk and Lk are conjugate Definition: The group Lk (or more precisely its conjugacy class) in 92. 1 is called the algebraic hull of a.
9.2.2
A useful example is the following
[Zimmer 4]: Suppose a : S x G ----" H"" where H is a connected fR.-group. If the G action on S is amenable, then the algebraic hull of a is amenable
9.2.3 Theorem
Proof: Let N be the radical of H, so that Q = H/N semisimple.. Let P be a minimal fR.-parabolic subgroup of Q. Then by amenability there is an a-invariant function S ----" M(Q""/P""), the latter being the space of probability measures on Q""/P"" . The action of Q"" on M(Q""/P"") is smooth by Corollary 3.2. 1 7 and the stabilizers are amenable and algebraic by Corollary 3..2..22. Since N"" acts trivially on Q""/P"", the H""-action on M(Q""/P"") is obviously also smooth, and since N"" is amenable, the stabilizer in H"" will also be amenable (4. L6). The result now follows from the cocycle reduction lemma (5..2J 1). We remark that virtually the same proof shows (using the fact that any connected group has a normal compact subgroup with a Lie quotient): .
Corollary: Let a : S x G ----" H where H is connected and the G action on S is amenable. Then a is equivalent to a cocycle into an amenable subgroup of H.
9.2.4
We also remark that if H is discrete, this result is no longer true. For a counter example, see [Zimmer 4]. When 9 ..24 is applied to the cocycle a: S x G ----" G, a(s, g) = g, where G is connected, it provides the basic step in the proof of the following result.
9.2.5 Theorem: Suppose G is connected and that S is an amenable ergodic G space. Then S is induced from an action of an amenable subgroup of G.
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Proof: Applying 9 ..2.4 to a(s,
g) = g, we can deduce from Example 4.2 1 8 that
there is a G-map S --+ G/H for some amenable subgroup H One can show that, quite generally, this implies that S is induced from an H-action. We refer the reader to [Zimmer 4] for details We recall that for free actions, amenability is an invariant of orbit equivalence (4 3 . 1 0) On the other hand, the condition that an action be induced from a certain subgroup, or from a certain type of subgroup, is a feature that depends on the action, not only on the action up to orbit equivalence. Thus, Theorem 9.2..5 has in common with the rigidity theorem for actions (5..2.1 ) that it deduces structural results about the action from features of orbit equivalence . We also recall that in 9.2. .5 (in contrast to the rigidity theorem) the actions will not be actions with finite invariant measure (4.3 . 3). We also remark that one of the hypotheses in the superrigidity theorem for cocycles (5.2 ..5) is actually a hypothesis about the algebraic hull of a cocycle. Namely the assumption that a is not equivalent to a cocycle into a subgroup of the form Lk where L c H is a proper algebraic k-grotip is of course j ust the assumption that the algebraic hull of a is in fact Hk. We will need the following result in Section 9.4. G --+ H k where H is a k-group and suppose the alge 0 braic hull of a is Hk Let H be the connected component of H. Then the skew product action (see the discussion preceding 4..2..2 1 ) of G on X = S x aHk/(H 0 )k is ergodic. Further, if we let a : X x G --+ Hk be a(s, [h], g) = a(s, g), then the algebraic hull of a is (H 0 )k 9.2.6
Proposition: Let a : S
x
A c X be an invariant set of positive measure . For each s E S, let A, = { y E Hk/(H 0 )k l (s, y) E A} Then A, is not empty on a set of positive measure, and since A is invariant under the skew product action, for each g E G, A, a(s, g) = A,g. By ergodicity of G on S, A, is not empty on a conull set of s, and so s --+ A, is an a-invariant map into the set of non-empty finite subsets of Hk/(H 0 )k By the cocycle reduction lemma (5 ..2. 1 1 ), it follows that a is equivalent to a cocycle into a subgroup of Hk leaving a finite subset F c Hk/(H 0 )k invariant, and such that A, is in the Hk-orbit of F for almost all s . However, since the algebraic hull of a is Hk, it follows that F = Hk/(H 0 )k Thus, A, = Hk/(H 0 )k a. e., so that A is conulL This establishes ergodicity . Essentially by definition, the projection map X --+ Hk/(H 0 )k onto the second coordinate is an a-invariant map. Thus by 4.2. 1 8(b), a is equivalent to a cocycle into (H 0 )k. Therefore, it only remains to prove the assertion that a is not equivalent to a cocycle into Lk, Proof: Let
..
.
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L c H 0 a k-subgroup . *
Suppose a is equivalent to a cocycle into Lk Then (4 2. 1 8(b)), there is an a-invariant map cp : S x Hk/(H 0 )k � Hk/Lk, i e , for each g E G, we have
cp(s, y)a(s, g) = cp((s, y) g) = cp(sg, ya(s, g)) for almost all (s, y) Let E = F(Hk/(H 0 )k, Hk/Lk) be the space of functions, so that Hk acts on E by (f h)( y) = f( yh - 1 ) h Furthermore, this action is smooth. (To see this, first observe that we can identify E as a subset of the space of all subsets of V = Hk/(H 0 h x Hk/Lk of cardinality d, where d is the order of Hk/(H 0 )k Thus to see that Hk acts smoothly on E, it suffices to see that Hk acts smoothly on Vd/S(d) where S(d) is the symmetric group on d letters. However, Hk is smooth on Vd by 1 13, i e , Hk\ Vd is countably separated.. Hence, so is Hk\( Vd/S(d) ) = (Hk\ V d)/(S(d) by 2 1 . 2 1.) Define c.D : S � E by (c.D(s))( y) = cp(s, y). Then the above consequence of a-invariance implies that c.I>(sg) = c.D(s) a(s, g).. In other words, c.D is an a-invariant function. As we have seen that Hk is smooth on E, the cocycle reduction lemma (5.. 2 . 1 1) applies and a � f3 where f3 takes values in the stabilizer of some element of E However, any such stabilizer clearly leaves a finite subset of H k/Lk invariant On the other hand, the stabilizer is the set of k-points of a k-group, and since the algebraic hull of a is H k, the stabilizer m ust actually be Hk Therefore, we deduce that Hk actually leaves a finite subset of Hk/Lk invariant, i e , Hk/Lk is finite. Thus, L :::J H 0, completing the proof
9.3
Actions of lattices and product actions
The rigidity theorem for ergodic actions (5..21) is one major respect in which ergodic actions of semisimple groups exhibit behavior markedly different from actions of amenable groups. The cohomology result (9J J) is another example of a marked difference between the semisimple and amenable cases In this section we wish to describe another such result
9.3.1 Theorem [Zimmer 1 3]: Let G be the real points of a connected simple algebraic fR-group, and r c G a lattice. Let S be an essentiallyfree ergodic r-space with finite invariant measure. Suppose, for i = 1 , 2, that C are discrete groups acting ergodically, with finite invariant measure, on (S;, Jl;), and that the r-action on S is orbit equivalent to the r 1 X r 2 -action on S 1 X S 2 Then either S 1 Or S 2 is .fi nite (modulo null sets).
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We have stated this for real groups but the same result is true if IR is replaced by any local field of characteristic 0 A similar type of result is also true for actions of G itself, as well as for certain actions of irreducible lattices in semi simple groups and for actions of certain discrete groups arising as fundamental groups of manifolds of negative curvature. We refer the reader to [Zimmer 1 3] for these and other generalizations. We also observe that this result stands in sharp contrast to the case when r is amenable . In fact, by 4.. 312 and 4.. 316, if r is amenable and acts properly ergodically and with finite invariant measure on S, then r is orbit equivalent to the r X f-action on s X S. As with many other results we have presented, our proof of 9.. 3. 1 will depend upon the behavior of measures on the variety G/P, where P is a minimal parabolic subgroup. In the proof of Theorem 9.31, we will need the following general result
r is a discrete group acting properly ergodically with invariant probability measure on (S, Jl.).. Then there is a discrete amenable group H which acts ergodically on S, preserves Jl, and such that for (almost) all s E S, sH c sf . 9.3.2
Proposition: Suppose
Proof: Since all our groups are countable, there is no problem in ignoring null
sets and we shall do so throughout the proof We first claim the following.. Suppose A, B c S where A, B are measurable disjoint sets and Jl.(A) = Jl(B) Then there exists a measure space isomorphism T: A -+ B such that for all x E A, x and Tx lie in the same f-orbit To prove this we define a sequence of subsets AN c A, BN c B inductively as follows . Let A 1 , B1 = 0. For N � 2, assume A 1 , . .. , AN - 1 , B 1 , .. , BN - 1 have been defined. Consider subsets D of A - A N - 1 of measure at least 1/N for which there is a measure space isomorphism T: D -+ E where E c B BN- 1 , with x and Tx in the same f-orbit. For a fixed N, there is clearly a maximal finite number of such triples (Di> Ti Ei) for which all Di> Ei are mutually disjoint Choose such a maximal collection of triples and let AN - 1 = uDi> BN - 1 = uEr Define AN = AN - 1 u AN - 1 , BN = B N - 1 u BN - 1 · Let A oo = u AN, Boo = uBN . By our construction of AN, BN, we clearly have the existence of a measure space isomorphism T: A ro -+ Boo with x, Tx in the same f-orbit If Aro = A, then we are done If not, let D c A - Aro with Jl.(D) > 0. Since Tis measure preserving Bro #- B, and we can choose E c B - B ro of positive measure . Since r acts ergodically D · r => E, and hence there is a subset Do c D of positive measure and y E r such that Do y c E However, if N is such that 1/N < J1.(D0), this contradicts the maximality of AN This shows A = A ro , verifying our assertion . Before proceeding, we introduce some terminology. If A c S is measurable, P = { B 1 , . . , B, } is a partition of A into disjoint measurable sets B; of equal -
>
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171
measure, and F i s a finite group o f measure space isomorphisms of A onto itself, let us say that F is ?-admissible if: for all TE F and all i, TB; = Bi for some j; F acts simply transitively on the elements of P; and for all x E A and TE F, x and Tx are the same r -orbit. The assertion of the preceding paragraph implies that for any partition P of A c S into r sets of equal measure there is a Z/rZ-action on A which is ?-admissible. If D c A is a measurable set and P is a partition of A, we let d(D, P) = inf {,u(D�B) / B is a union (possibly empty) of subsets belonging to P}. We remark that if D, E c A, then d(D,
P) � d(E, P) + ,u(D� E).
Now let {A;} be a sequence of measurable subsets of S which are dense in the measure algebra of S, and such that for each i, A; = A n for infinitely many n > i. We claim that we can inductively construct for each i the following: ( 1 ) a partition P; of S into 2"' mutually disjoint measurable subsets of equal measure, for some n; � i ; (2) a ?;-admissible action on S of a finite abelian group F;; such that (3) P; + 1 ::::J P; (i . e. P; + 1 is a "finer" partition); (4) F; + 1 ::::J F;; (.5) d(A;, P;) < 1/2 ; For i = 1 , any partition into two sets of measure 1/2 suffices . We now construct P;+ 1 from P; Let P; = { D 1 , , D 2 n.} We can find a number m > n; and for eachj a partition Qi= {Cid of Di such that: (6) ,u( C ik) = lj2m for all j, k; and (7) for each j, d(A ; + 1 n D h Qi) < 2 - ( i + Z + n,) We can then choose a partition R of D1 such that: (8) for some n; + 1 > m every set in R has measure 2 - (n, + ! l ; and (9) if Ti E F; is the unique element with Ti(D t ) = Dh then d(Tj 1 (Cik), R ) < 2 - (i + Z + n , + m) for any j, k. Define P; + 1 to be the partition of S into sets of the form T(E), where E E R and TE F;. Since F; is measure preserving, P; + 1 is a partition of S into 2"' + ' sets of equal measure. It follows from (9) that for all j, k, d( Cik, P; + t ) < 2 - U + 2 + n , + ml, and hence for any set E c D i which is a disjoint union of elements of Qi that
From this inequality and (7) we deduce that for all j,
Thus d(A ; + 1 , P; + t ) < 2"'2 - (i + 1 + n,) =
2 - (i + l J
This establishes condition (.5). To
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complete the induction, we construct a P; + 1 -admissible finite abelian group F; + 1 acting on S such that (4) is also satisfied. By our earlier remarks there is an R-admissible action of Z/rZ on D 1 where r = 2"' + ' - "' Let X E Z/rZ, j > 1, and Ti E F; be as in (9). Let X act on Di by Ti)Jj 1 This extends the ZjrZ action to all of S and clearly Z/rZ commutes with F;. Thus, letting F i + 1 be the group generated by Z/rZ and F;, the induction argument is complete Let H = u F; . Then H is abelian and hence amenable . Furthermore sH c sr and so to prove the proposition it suffices to see that H acts ergodically on S. Suppose A c S is H-invariant Since F; c H transitively permutes the elements of P;, it follows for B, C E P;, we have .u(A n B) = .u(A n C).. Therefore ,u(A n B) = 2 - n, .u(A) = ,u(B).u(A).. It follows that .u(A n B) = .u(B),u(A) for any set B which is a union of elements of uP; . From (5) and the fact that A; = An for infinitely many n > i, we deduce that for each i, .u(A n A ;) = .u(A;),u(A).. Since { A;} is dense in the measure algebra of S, we obtain .u(A n B) = .u(B),u(A) for all measurable B. Hence, letting B = A, we conclude that A is either null or conull, establishing ergodicity. We now turn to the proof of 9 J J .
Proof o f Theorem 9.3.1 : Throughout this proof� "algebraic group" will mean the set of real points of an algebraic IR-group. If S; is not finite, then 1; acts properly ergodically on S;. Choose an amenable subgroup A; acting on S; as in 9. 3.2, i e ' with xA; c xr; for X E S; Let 8: s --+ St X s2 be the orbit equivalence of the r-action with the r 1 X r 2 action . Let ex : (St X S2) X r 1 X r 2 --+ r c G be the cocycle corresponding to the orbit equivalence 8 - l (Example 4 .2..8), i.e. , for X E s X Sz, h E r 1 X r 2 , 8 - 1 (x)ex(x, h) = 8 - 1 (xh). By Theorem 9.2.. 3 (and Example 1 4. 2. 1 8), there is an amenable algebraic subgroup H c G and an ex I S 1 x S 2 x A t x Arinvariant function cp : S 1 x S2 --+ GjH. As the following lemma shows, to provide the desired contradiction, it actually suffices to show that there is such an ex-invariant function.. .
Lemma: To prove the theorem, it suffices to show that there is a proper algebraic subgroup L c G and an ex-invariant function cp : S 1 x S2 --+ GjL.
9.3.3
Let "' ; s --+ GjL be "' = cp 8.. For s E S, ')! E r, h E r 1 X r 2 such that 8(s) h = 8(sy), we have by the definition of ex that ex(8(s), h) = y, and hence that
Proof:
0
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Further results on ergodic actions
tjf(sy) = cp(B(sy)) = cp(B(s) h) = cp(B(s)) ex(B(s), h) [by ex-invariance of cp]. = t/J (s) Y In other words, t/1 : S --+ G/L is a 1 -map . If J1 is the 1 -invariant measure on S, then t/J*(Jl) will be a [-invariant probability measure on G/L We can then define a G-invariant probability measure on G/L as follows. Let m be the G-invariant measure on G/1 For A c G/L measurable, (t/J *Jl)(Ag - t ) depends only on the image of g in G/f, and thus we can define v(A) = j (t/J*(Jl))(Ag- t )dm. This is then clearly a G-in variant probability measure on G/L However, since L is a proper algebraic subgroup, this is impossible by the Borel density theorem (3.2..5) . This proves the lemma. Therefore, we need to pass from the existence of an a i St x S 2 x At x A 2 invariant function to the existence of a suitable ex-invariant function. We shall do this by developing a uniqueness property for a ! S t x S 2 x A t x A z-invariant maps. Suppose that for each s E S t we have an algebraic subgroup H, c G such that {(s, g)j gE H,} is a measurable subset of St x G. Let p, : G --+ G/H, be the natural projection. Let cp : S 1 x S 2 --+ G be a Borel map. Let us say that ( cp, { H,}) is ex-admissible if for almost all s E S 1 , p, o cp is an ex ! { s } x S 2 x A z-invariant function . For s E S 1 , h E 1 1 , x E S 2 , and a E A 2 , we have from the cocycle relation that
ex(s, x, h, e)ex(sh, x, e, a) = ex(s,
x,
e, a)a(s, xa, h, e). For a fixed h E 1 1 , and ex-admissible (cp, {H,}), define h *(cp, {H,}) to be the pair (t/1, fi,) where t/J(s, x) = cp(s · h, x) · ex(sh, x, h - t , e) and fi, = Hsh· The above consequence of the cocycle relation is readily seen to imply that h * (cp, H,) is also ex-admissible for any h E 1 1 Furthermore, it is clear that if (cp i , { H�}), i = 1, 2, are ex-admissible, and S 1 = B u C is a di�joint decomposition into measurable sets, then defining cp to be ({J 1 on B x S 2 and q>z on C x Sz, and
defining H, similarly from H �, we still obtain an ex-admissible pair. It follows from these remarks and ergodicity of the r 1 -action on S 1 that there exists an a-admissible (cp, {H,}) such that dim H, is essentially constant over s E S 1 and that for any admissible (t/1, { J,}), we have dim H, � dim J, a.e. The following observation is basic to the proof of Theorem 9.3. 1 .
Lemma: Suppose (cp, {H,}), (t/1, {J,}) are ex-admissible and both satisfy the above minimality of dimension property.. Then (i) H? and J ? are conjugate for almost all s.. (ii) The conjugacy class of H ? is essentially constant in s. In particular, we can 9.3.4
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choose H, satisfying the minimality of dimension property with H ? = H 0, independent of s. (iii) With such a choice, let p : G -+ GjN(H 0), where N(H 0) is the normalizer of H 0 (and is therefore algebraic). Suppose we also have J ? = H 0 a e , so that H ,, ], c N(H 0). Then p o cp = p o lj; Proof: Consider the map (cp, lj;): s 1 X s 2 -+ G For each s E Sj, let p, : G -+ GjH,, q, : G -+ GjJ, be the projections. Then (p, o q;, q, o rf;) yields an a l {s} X s2 X A 2invariant map {s} x S 2 -+ GjH , x Gjl, . Since A 2 is ergodic on S2, and G-orbits are locally closed in G/H, x G/1, (3. 1.3), it follows from the cocycle reduc tion lemma (5 ..21 1 ) and Example 4.2. 1 8(b), as in the proof of 9.2J, that (p, o cp, q, o lj;)({s} x S 2 ) is (a.e.) contained in a single G-orbit in GjH, x GjJ,. The stabilizer of such an orbit can be chosen to be of the form L, = H, n bJ ,b- 1 for some b E G. It is a technical exercise to see that s -+ L, can be chosen measur ably, and hence ((cp, lj;), {L,} ) is also a-admissible. By the dimension property, dim L, = dim H, = dim (J,) a. e. , and hence H ? and J ? are conjugate. To see that the conjugacy class of H ? is independent of s E S 1 , observe first that if h E r 1 , h * (cp, { H,}) will also satisfy the minimality of dimension condition, and so by the above H? and H ?h are conjugate. We now employ an argument of [Auslander-Moore 1]. Let L(H,) denote the Lie algebra of H,, so that L(H,) c L( G) is a linear subspace, and the dimension of this subspace is essentially constant for s E S 1 , say of dimension d. Since H ? and H?h are conjugate, it follows that L(H,) and L(H,h) are in the same Ad( G)-orbit for the action of Ad( G) on the Grassmann variety V of d-planes in L(G). However, the action of Ad( G) on V is smooth (3.. 13), and thus V/Ad(G) is countably separated.. Since L(H,) and L(H,h) are equal when projected to V/(Ad(G)), the map s -+ (pr�jection of L(H ,) in V/Ad( G)) is an essentially r � -invariant function on an ergodic r 1 -space taking values in a countably separated Bore! space. It follows that this map is essentially constant, or equivalently, on a conull set in S 1 , all L(H,) are in the same Ad(G)-orbit This implies that on a conull set in S 1 , all H ? are conjugate in G . The usual correspondence of cosets of a subgroup and that of a conjugate subgroup then enable us to choose an admissible (cp, {H,} ) satis(ying the minimality condition with H ? = H 0 a. e. Finally, suppose we also have J ? = H 0 a.e. As above, the image of the map (cp, lj;) will lie in a single G-orbit, with the stabilizer of a point in the orbit of the form L, = H, n bJ,b - 1 , for some b E G Since dim H, = dim J, = dim L,, bH 0 b - 1 = H 0, i.e., b E N(H 0 ).. Thus the image ofthe relevant orbit of GjH, x GjJ, in G/N(H 0 ) x GjN(H 0 ) contains a point on the diagonal, and hence is contained in the diagonal. Therefore, the image of (p o cp, p o lj;) is essentially contained in the diagonal, and this proves the final assertion of the lemma.
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9.3.5
Lemma:
Let H, as in the conclusion of Lemma 9. 3.4. Then H0 i= { e } .
Proof: If it were, each H, would be finite Fix s E S 1 so that all required condi tions which are known to hold a.e . hold on almost all { s } x S2 In particular, qy , : S2 ---+ G/H, given by qy,(x) (p, o qy)(s, x) is an a j { s} x S2 x A z-invariant function. We can choose a compact set B c GjH, such that
We now complete the proof of Theorem 9 . 1 1 As in Lemma 9.3.4 choose (qy, {H,}) a-admissible, satisfying the minimality of dimension condition, and H� = H 0 As above, let h E f1 As observed preceding Lemma 9. 3.4, h*(qy, {H,}) will also be a-admissible It follows from the final conclusion of Lemma 9.3.4 and the definition of h*(qy, {H,}) that p(qy(sh, x)) a(sh, x, h, e)) = p(qy(s, x) ).. In other words, p o qy : S1 x S2 ---+ G/N(H0) is an a j S1 x { x} x f 1 -invariant map for a. e. x E S2 By our construction at the beginning of the proof, H 0 =1- G, and by Lemma 9. 1 5, H 0 i= { e } , and so N(H0) =1- G.. We can now repeat our entire argument working with a I S 1 x { x} x r 1 -invariant functions . Lemmas 9.3 . 4 and 9.3..5 hold in this situation as well, and repeating the arguments immediately above using }' E f 2 rather than h E f 1 , we deduce the existence of an a-invariant map qy : S 1 x S2 ---+ G/L for some non-trivial proper algebraic subgroup L c G. By Lemma 9.3.3, this completes the proof. 9.4
Rigidity and entropy
In Section 52 we applied the superrigidity theorem for cocycles (5. 2..5) to the cocycles arising from an orbit equivalence to deduce the rigidity theorem for ergodic actions . Theorem 5.2 ..5 can also be applied to the derivative cocycle for an action on a manifold (Example 4.2.3) or to other associated geometrically defined cocycles . This is a basic step in proving results about actions of semi simple groups or their lattices on compact manifolds . The full proofs, however, require some significant geometric background and so we shall not present these results here.. As an introduction to this subject, we shall present an application of superrigidity to the computation of entropy, a basic measure theoretic invariant of the diffeomorphisms defined by single elements of the
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acting group. This particular consequence of superrigidity was first observed in [F urstenberg 5]. For further developments of the applications of superrigidity to actions on manifolds, we refer the reader to [Zimmer 1 5, 1 8, 1 9].. We shall first review the concept of entropy for an integer action . We will not provide proofs, but refer the reader to [Billingsley 1] and [Ornstein 2].. We assume that (S, Jl) is a Z-space, where J1 is a Z-invariant probability measure. Let T: S --+ S be given by Ts = s 1 . To this transformation, we will associate a non-negative real number h(T), called the (Kolmogorov-Sinai) entropy of T We begin by seeking a quantitative measure of randomness. Namely, suppose p = (p 1 , . . , pk ) is a probability measure on { 1 , . , k}, ie , 0 ;;:; p; ;;:; 1, 'L p ; = 1 . W e seek a number H(p), which we want t o measure the randomness of p, satisfying the following axioms: (i) H is continuous in p, H(p) � 0, H is symmetric, and H(p, 0) = H(p) (ii) H(l, 0, , 0) = 0. (No randomness.) (iii) For each k, H(p1, . , Pk ) achieves a maximum at p; = 1 /k for all i. (iv) If (p 1 , , Pk ), (q 1 , . . , qk ) are probability vectors, then We then have:
9.4.1
Theorem
[Khinchin 1]: Up to a scalar multiple, the only function satisfying
(i}-{iv) is H(p 1 , . . , Pk) =
- 'L
p ; log p ;.
H( p) is called the entropy of p We now let (S, Jl) be a standard measure space, J.l(S) = 1 . By a partition d of S we mean a collection d = { A 1 , . . . , Ak }, where A; c S are mutually disjoint measurable sets and u A; = S Define H(d) = H(JL(Al ), . . , Jl(Ak ) ). If d, f!J are two partitions, let d v !!4 be the partition d v !!4 = {A; n Bi }. It is then easy to check from property (iv) above that H(d) ;;:; H(d v !!4) ;;:; H(d) + H(f!J). If d = !!4, then clearly H(d v !!4) = H(d). Hence if !!4 is close to d, H(d v !!4) will be close to H(d).. On the other hand, suppose d and !!4 are independent partitions, i e , Jl(A; n Bi) = Jl(A;)Jl(B il (for example, partitions from partitions in factors of a product space.) Then we have H(d v !!4) = H(d) + H(f!B). Recall the map T: S --+ S The entropy of T will measure the rate of growth of H(d V T(d) V . . V r(d))..
1 77
Further results on ergodic actions
9.4.2 Definition: (i) Set h( T, d) = lim H(d V T(d) V . . . V T" - 1 (d))/n. (ii) Set h( T) = sup {h( T, d) I A is a partition of S}. h(T) is called the entropy of T (or of the Z-action), and is clearly an invariant of the Z-action up to conjugacy . One can compute h( T) because of the following:
(Sinai): If d is a generating partition for the Z-action, i.e., the smallest closed Boolean subalgebra of the measure algebra B(S) that contains d and is Z-invariant is B(S) itself, then h(T) = h(T, d)..
9.4.3
Theorem
9.4.4
Example:
(a) Let (X, p) be a finite probability space. Let Q
=
00
I (X, p),
- oo
and define T: Q --> Q by (Tw)n = Wn + 1 This is called the Bernoulli shift o n the state space (X, p).. Then h(T) = H(p).. (In particular, h(T) can take any positive value. ) (b) If T is a rotation of the circle, then h(T) = 0 (c) Let A c SL(2, lP) be the positive diagonal matrices, and r c SL(2, lP) a cocompact lattice. For a E A, let h(a) be the entropy of a acting on SL(2, IP)jr Then up to scale, h(a) = log 1 1 a 1 1 . Thus, h(T) can take any non-negative value for T a diffeomorphism of a compact manifold. We now state two fundamental results that describe the relationship of entropy to Bernoulli shifts .
[Ornstein 1]: Any two Bernoulli shifts with the same entropy define conjugate Z-actions. 9.4.5
Theorem
[Sinai 1]: Suppose S is a Z-space with entropy h. Then there is a Bernoulli shift with entropy h, say on Q, and a measure preserving Z-map S --> n
9.4.6 Theorem
For a C 2 -diffeomorphism, the entropy may be computed from properties of the derivative cocycle on the tangent bundle . To describe this result, we first describe some general results concerning the asymptotic behavior of cocycles of Z-actions. If A is any n x n matrix, then lim *log + 11 A" 11 exists and is equal to max {log + I A- l l
A is an eigenvalue of A} If (S, p) is a Z-space, and a : S x Z --> GL(n, C) then a(s, n) = a(s, 1 )a(s 1 , 1 ), . . , a(s · n - 1 , 1 ), and so is naturally a product of n (unequal in general) matrices . For cocycles which are sufficiently regular, one can obtain
1 78
Ergodic theory and semisimple groups
a similar result on the behavior oflog + ll a(s, n) 11 Namely, let us call a cocycle IX "tempered" if I l a( , n) 11 E L "'(S, fl) for each n.. Define (e(a) )(s) = lim * log + ll a(s, n) 11 n � eo
if it exists
Theorem [Furstenberg-Kesten 1]: Let (S, Jl) be a Z-space where f1 is an invariant probability measure. Suppose IX : S x Z ---> GL(n, C) is a tempered cocycle Then (i) e(a)(s) existsfor almost all s E A, and e(a) is essentially Z-invariant. Hence, if Z acts ergodically, e(a) is constant. (ii) If IX and [3 are tempered and IX � [3, then e(a)(s) = e([J)(s) (a. e.).
9.4.7
For a proof of 94..7, see [Kingman 1] and [Derriennic 1]. A basic result on the entropy of difleomorphisms is the following theorem of Pesin If the integers Z act on a compact manifold M by diffeomorphisms, compactness implies that 11 df� 11 is bounded over x E M for each fixed n (for any choice qf Riemannian metric) and hence that the derivative cocycle (Example 4..2..3) is tempered (since we can measurably trivialize the tangent bundle with orthonormal sections). Theorem [Pesin 1]: Suppose the integers act on a compact manifold by 2 C - diffeomorphisms so as to be volume preserving. Set Ts = s· L Let IX be the derivative cocycle, and let AP(a) be the p-th exterior power of IX Then the entropy
9.4.8
is given by h(T) = J[max e(N(a))(s)]ds In particular, in the ergodic case p
h(T) = max e(N(a)). p
Pesin's result depends in part upon a sharpening of 94. 7 due to [Oseledec 1] called the "multiplicative ergodic theorem" Before describing this, we make some remarks about a single matrix A E GL(n, IC) Let A;, i = 1 , . . . , k be the distinct real numbers of the form log + I ) I where A is an eigenvalue of A, and assume A1 < A2 .. < Ak Then there are subspaces 0 c V 1' c V2 c . . c Vk = IC" 0
*
such that for v E V; but v � V; - 1 , we have lim Mog + II A"v ll
=
Ai ··
Oseledec's
theorem generalizes this spectral type decomposition to tempered cocycles. Theorem [Oseledec 1]: Let (S, /1) and IX be as in 94.. 7 and assume the Z action is ergodic. Then there exist A 1 , . . , )ok E lP, A1 < ) 2 < . . . < Ak, positive integers 1 � n 1 < nz < < nk = n, and a-invariant measurable maps Vi : S ---> Grn,(C"), the
9.4.9
0
1 79
Further results on ergodic actions
latter being the Grassmann variety of n;-planes in en, with V;(s) c V; + 1 (s) such that for almost all s E S, we have for all v E V;(s), v i$ V; - 1 (s) lim � log+ ll o:(s, n)v ll = A; n - OCJ n Remark:
Clearly e(o:) = },k We also have max e(AP(a)) = �(n; - n; - 1 )).; The ),k p
are called characteristic (or Lyapunov) exponents of a. For a more recent proof of 9.4 . 9, see [Raghunathan 2] Example: If o:(s, n) = An where A E GL(n, C), then Theorem 94. 9 reduces to the observation we made following 9.4..8. If we know that a tempered cocycle a takes values in an algebraic subgroup of GL(n, C), we can say more about e(a). For example, if a takes values in a unipotent subgroup, then e(a) = 0 (as do all the characteristic exponents) . More generally, if G c GL(n, q is a connected algebraic subgroup, we can write G = R ex. U where U is the unipotent radical and R is reductive. For a cocycle a taking values in G, let o:R be the reductive component, i.e , ClR = p a where p : G � R is the projection. Then we have e(a) = e(o:R). We shall need the follow ing generalization of this remark 9.4.10
o
Proposition: Suppose a is a tempered cocycle taking values in GL(n, IR). Let H c G L(n, q be an IR-group. Suppose H is connected, and write H rK = R ex. U where R is reductive and U is unipotent. Let f3 be a cocycle equivalent to a with f3(S x Z) c HYK and, as above, let /3R be the composition of f3 with projection onto R. Finally, assume f3R is also tempered. Then e(f3R) = e(a). In concluding these preliminaries, we mention that the entropy and the characteristic exponents of a diffeomorphism have important consequences for the study of classical dynamical properties of the diffeomorphism, for instance periodic points . See [Bowen 1] and [Katok 1] for example . Now let G be a connected simple IR-group . Let G be the algebraic universal covering group, and n : GYK � SL(n, IR) an irreducible rational representation . If g 1 , g 2 E GrK and g 1 = g 2 in GrK, then g 1 = g 2 z where z E Z(GrK) (the center of GrK). Since Z(GrK) is finite, it follows that 9.4. 1 1
max { I A I I A an eigenvalue of n(g 1 ) } = max { I A I ! ). an eigenvalue of n(g 2)} .
1 80
Ergodic theory and semisimple groups
Thus, for g E G11, max { I X I I ), an eigenvalue of g} is well defined where g E G11 is any element projecting to g. We now present an application of the superrigidity theorem for cocycles (5 ..2..5). Combining 5.2 5 and 9. 4 . 8, we obtain the following 9.4. 1 2 Theorem: Let G be a connected almost simple fR-group with fR-rank(G) � 2. Suppose Gg acts by C 2 volume preserving diffeomorphisms on a compact manifold M, and suppose the action is ergodic. For g E G�, let h(g) denote the entropy of the diffeomorphism defined by the action of g on M. Then either h(g) = O for all g, or for each g E G�, there is an fR-rational representation n : G ---> SL(r, C) for some r, such that
h(g) = max {log(l A I) I A is an eigenvalue of n(g) } . Further, if n = dim M, n is a n exterior power of a representation a of G , where dim (J � n. Corollary: For a fixed n, and fixed g E G�, the set of possible values ofh(g) over all volume preserving, ergodic, C 2 -actions of G� on compact manifolds of dimension at most n is .finite. 9.4.1 3
Proof of 9.4.1 2:
Let a : M
x
G� ---> GL(n, fR) c GL(n, C) be the derivative cocycle.
For g E G� let a9 = a I M x { gn } . Thus h(g) = J max e(N(a9) ) by Pesin's theorem. p
Now let H11 c GL(n, fR) be the algebraic hull of a. As in 9.2.6, let S = M x aH11/(H 0)11, and define f3 = a as in 9.2. 6 as well. Since H11/�H 0)11 is finite, S has a finite invariant measure. Clearly for any p, e( N(a9) ) = e(AP(f39)). By 9 2. 6, the algebraic hull of f3 is (H 0)11. Let H0 = Lr>< U where L is reductive, U is unipotent, and L and U are fR-groups . Let y = f3L so that y : S x G� ---> L11 is a cocycle with algebraic hull L11 Let q : L11 ---> L11j[L, L]11. Then the algebraic hull of q y is LR/[L, L]11, and hence by Corollary 9. 1. 3 L11/[L, L]11 is compact Since L = [L, L] ·· Z(L) where Z(L) is the center of L, and [L, L] n Z(L) is finite, it follows that Z(L)11 is also compact We can write the fR-group L/Z(L) as a product of semisimple fR-groups L/Z(L) = L 1 x Lz such that (L2)11 is compact, and (Lt)R is center free and with no compact factors.. Let q l : L11 ---> (Lt)11 be the pr�jection. Then q1 y : S x G�-> (L 1 )11 is a cocycle with algebraic hull (Lt)R, and by the superrigidity theorem for cocycles (5.. 2 .5), there is a rational homomorphism n : G ---> L 1 defined over IR o
o
181
Further results on ergodic actions
such that q 1 y � IX, 1 a � where IX, i s the cocycle a,(s, g) = n(g). Thus, w e can write, for g E G2 and a. e. s E S, cp(s)q l (r(s, g))cp(sg) - 1 = n(g) where cp : S -> (L l )R is Bore! We can choose a Bore! section[ : (LdR -> LR of q 1 Then q 1 (f(cp(s))y(s, g)f(cp(sgW 1 ) = n(g). In other words, replacing }' by an equivalent cocycle 6 : S x G� --> LR, we have q 1 (6(s, g)) = n(g) We can consider n as a homomorphism n : G -> L 1 , where G is the (algebraic) universal covering group of G Then we can lift this to a homomorphism ii :G -> [L, L] c GL(n, C) defined over IR. Thus, for each g E G�, we have q 1 (6(s, g)) = q 1 (ii(g)) for almost all s, where {j E GR projects to g under the covering map. However, since (L2)R and Z(L2)11 are compact, (ker ql)R is compact Thus, ii(g) = b(s, g) b where b is some element of the compact normal subgroup (ker q l )R of LR It follows that for any g E G�, and any p, that APb9 is tempered and that e(AP69) = max {log i A. I I A. is an eigenvalue of AP(ii(g)) } o
Then
h(g) = J max e(AP1X9) = J max e(APf39) p
p
by earlier observations,
= J max e(APbp),by Proposition 94. 1 1 p
This completes the proof Theorem 94. 12 can in fact be extended to actions of lattices on manifolds. To see this, we first observe that the superrigidity theorem can be extended to actions of lattices by inducing. More precisely, we have the following 9.4.1 4 Theorem (A corollary of 5.2..5): Let r c G� be a lattice where G is a connected almost IF!-simple IF!-group with IF!-rank(G) � 2. Let S be an ergodic !-space with finite invariant measure. Let H be a connected IF!-slmple Lie group (trivial center) with HR not compact.. Suppose IX · S x r -> H11 is a cocycle with algebraic hull HR. Then there is an IF!-rational homomorphism n . G ---> H such that IX � 1Xni f . Proof: Let f : G�j[' x G� -> r be a strict cocycle corresponding to the identity map 1 -> r (4..2. 1 3). Define the G�-action induced from the r -action on S, X = G�/1 x 1 S (4. 2.21). Define a cocycle {J : X x G� -> HR by {J((y, s), g) = IX{s, f(y, g)).. Then X has a finite G�-invariant measure, and the algebraic hull of f3 is H11 . To see this last assertion, assume that f3 is equivalent to a cocycle with algebraic hull L11 c H11, i e , there is a {3-invariant function cp : X -> H11/L11 (4.2. 1 8(b)). Let r act on F(S, H11/ L11) by the a-twisted action. Under the identifi-
Ergodic theory and semisimple groups
1 82
cation F(X , H'R/L"') with F(G�jr, F(S, H'R/LrK)), cp corresponds to an f-invariant element of F(G�jr, F(S, H'R/L'R)).. By (4. 2. 1 9), there is a r-irivariant element in F(S H'R/ L'f<), i.e. , an a-invariant function S ----+ H'R/L'R·· Hence from 4.2J 8(b), we deduce that L"' = H 'R since the algebraic hull of a is H'R This verifies that the algebraic hull of f3 is also H'R Thus, we can apply 5.2 . 5 and deduce that f3 � a" where n is an IR-rational homomorphism n : G ----+ H. We claim that a � a rrlr If we knew that f3 was a strict cocycle and that f3 and a" were strictly equivalent, then simply by writing out the equation for the equivalence we could deduce a � a rr1r To deal with the almost everywhere conditions, we will reformulate matters so as to be able to apply (4.2. 1 9).. Let H x H act on H by (a, b) h = ahb - 1 . Then if a,/3 are cocycles on some space taking values in H'R, then they are equivalent if and only if there is an (a, /3)-invariant function into H'R, where (a, /3) is an HP. x H'J1-valued cocycle. Let n : X x G� -+ HrK be the cocycle n((y, s), g) = n(f(s, g)).. Since f(s, g) corre sponds to the identity r ----+ r, f(s, g) = y(s)gy(sg) 1 and hence it follows that n � a" Thus, since we also have a" � /3, it follows that there exists a (/3, n)-invariant function lj; : X ----+ H'J1· Once again, under the identification of F(X, H'R) with F(G�jr, F(S, HP.)), lj; corresponds to an ./-invariant function G�jr ----+ F(S, H"') where r acts on F(S, H"') by (y h)(s) = a(s, y)h(sy)n(y)- 1 • By (4.. 2. 1 9), there is a r-invariant function under this r-action, Le., there is hEF(S, H'R) such that h(s) = a(s, y)h(sy)n(y) - 1 But this clearly implies that a � a rrlr , completing the proof. Exactly the same proof as the proof of theorem 9.4. 1 2 now yields the following. ,
-
,
Theorem: Let G be a connected almost IR-simple IR-group with IR rank(G) � 2 . Let r c G� be a lattice. Suppose r acts by C 2 , volume-preserving diffeomorphisms on a compact manifold M, and suppose the action is ergodic. For y E r, let h(y) denote the entropy of the diffeomorphism defined by the action of y on M Then either h(y) = 0 for all y E r, or for each y there is an IR-rational representation n . G -+ SL(r, C) for some r, such that h(y) = max{log( I A. I ) I A. is an eigenvalue of n(Y) }. Further, if n = dim M, then n is an exterior power of a representation a of G where dim a � n. 9.4.1 5
.
..
9.4�1 6
Corollary:
The conclusion of 9A. l 3 is true for actions of r.
Theorem 9.4. 1 5 puts strong restrictions on the type of actions r can have on low dimensional manifolds. For example:
Further results on ergodic actions
1 83
9.4.1 7 Corollary: For n � 3, let SL(n, Z ) act ergodically by volume preserving C 2 -diffeomorphisms of a compact manifold M. Suppose dim M � n - L Then h(y) = 0 for all y E SL(n, Z )..
SL(n, IR) has no non-trivial rational representation of dimension smaller than n 941 .5 then implies the result For another result of this type, see 10. L1 0.
Proof:
9.5
Complements
In this section we indicate without proof some further results concerning lattices and ergodic actions. We make no attempt at a survey but restrict ourselves to some results which directly complement those of the previous chapters.
A
Ergodicity of restriction
Let G = llG; where G; is a simple, connected, non-compact Lie group with finite center, and let r c G be an irreducible lattice. Let (S, f.l) be an ergodic G-space, where f.1 is a quasi-invariant measure. The question arises as to when the restriction of the G-action to r will still be ergodic. If the G-action is essentially transitive, Moore's theorems (chapter 2) provide us with a complete answer . Namely, if S = G/G0, then r is ergodic on S if and only if Go is not compact Therefore, it remains only to consider the properly ergodic case 9.5.1. Theorem [Zimmer 2]: Suppose G, r as above, and that S is a properly ergodic G-space. Then the restriction of the G-action to r is still ergodic.
B
Dense orbits
In Chapter 2 we established a number of results concerning ergodicity of the action of H 1 on G/H2 , where H; c C are closed subgroups. In this case, ergodicity implies that almost every orbit is dense (2. 1.7).. The question then arises as to when one can deduce that every orbit is dense, or more generally, to identify the dense orbits. A group action on a topological space is called minimal if every orbit is dense. In analogy with 2.2.3, we have the following.
1 84
Ergodic theory and semisimple groups
Proposition: Let H 1 , H 2 c G be closed subgroups of a locally compact group. Then H 1 acts minimally on G/H2 if and only if H 2 acts minimally on G/H 1 .
9.5.2
A general minimality assertion is the following.
[Mostow 2]: Let G be a connected semisimple Lie group, P c G a parabolic subgroup and r c G a lattice . Then r acts minimally on G/P.
9.5.3 Theorem
If r is a uniform lattice, we have the following stronger assertion. Theorem [ Veech 1]: Let G be (the real points of) a semisimple IR-algebraic group without compact factors and let N c G be the unipotent radical of a minimal parabolic subgroup. If r c G 0 is a uniform lattice, then r acts minimally on G 0/N. 9.5.4
For G = SL(2, IR), this result was first proved by [Hedlund 1] Veech in fact proves a stronger theorem which in particular, when applied to SL(n, IR) yields the following. By a p-frame in IR" we mean an ordered linearly independent set in IR" with p elements . Theorem [ Veech 1]: JfT c SL(n, IR) is a uniform lattice, then for each p, 1 � p � n - 1, r acts minimally on the space of p-frames.
9.5.5
For non-uniform lattices, 9.5.5 is of course no longer true. We call a p-frame irrational if its IR-linear span contains no elements of
[Dani-Raghavanl]: For 1 � p � n - 1, the orbit of any irrational pjrame under SL(n, Z ) is dense in the space of all pjrames . 9.5.6
Theorem
Questions involving minimality of actions and identifying minimal invariant (closed) subsets is closely related to the question of finding all locally finite invariant measures which are ergodic under the action. The reader is referred to [Furstenberg 3], [Veech 2], and the extensive work of [Dani 1-3] for results in this direction
Further results on ergodic actions
C
!85
Ratner's Rigidity Theorem
The rigidity results we have considered in Chapter 5 do not apply to a pair of subgroups in pSL(2, IR). In this case, we have the following theorem Theorem [Ratner 1]: Let N c PSL(2, IR) be the ! -parameter upper tri angular unipotent group. Suppose r 1 . r 2 c PSL(2, IR) are lattices. If the N-actions on PSL(2, IR)jr 1 and PSL(2, IR)/r2 are conjugate, then r 1 and r2 are conjugate subgroups 9.5.7
D
Orbit equivalence
Let M be a complete manifold of finite volume and constant negative sectional curvature. Then r = n 1 (M ) can be realized as a lattice subgroup ofO( l , n). Hence, if r' c G is a lattice where G is simple and IR-rank(G) � 2, the results of section 5.2 imply that r and r' do not have orbit equivalent free ergodic actions with finite invariant measure. If we only assume that the curvature of M is negative and bounded away from 0, then we do not know that r is a lattice in 0(1 , n) and hence section 5.2 does not apply. However, the non-orbit equivalence assertion is still valid. Theorem [Zimmer 10]: Let M be a complete manifold of finite volume and negative sectional curvature bounded away from 0.. Let r c n 1 (M ) be any sub group, and let r' be as above. Then r and r' do not have orbit equivalent free ergodic actions with .fi nite invariant measure. 9.5.8
The proof of this result depends on the following generalization of Theorem 81 .3.. The essential ideas of the proof are similar to those of the proof of 8. 1.1
[Zimmer 10]: Let S be an irreducible ergodic G-space with finite invariant measure, where G is a connected semisimple Lie group with finite center, no compact factors, and IR-rank at least 2. Let P c G be a parabolic st�bgroup. Suppose X is a G-space with finite invariant measurefor which we have measurable G-maps S x G/P -+ X -+ S whose composition is the projection. Then X is of the form X = S x G/P' where P' :::J P, and the maps are the natural ones. 9.5.9
E
Theorem
Topological quotients of r -actions
The topological version of Theorem 8 . 13 is also true. Namely:
186
Ergodic theory and semisimple groups
9.5.1 0 Theorem [Dani 4]: Let G be a connected semisimple Lie group with finite center and IR-rank(G) � 2 . Let P c G be a parabolic subgroup and r c G an irreducible lattice. Suppose X is a compact Hausdorff r-space and there is a continuous surjective r-map
This was first established in the special case G = SL(n, IR), G/P = IP"- 1 , and r commensurable with SL(n, Z )(n � 3) in [Zimmer 1 1], using 95 6. A counter example to 95. 10 for lattices in groups of IR-rank one has been given by [Spatzier 1].
Generalizations to p-adic groups and S-arithmetic groups
10
In this chapter we indicate how to extend some of the results of the preceding chapters to lattices and actions of groups which are (finite) products of semi simple groups over possibly varying local fields . More precisely, for each prime p E Z let CQ P be the field of p-adic numbers, and let CQ ro = IR Let V= {primes in Z } u { oo }, and S c V a finite subset Suppose that for each p E S, Gp is a connected, semisimple CQp-group such that (Gp)([)p has no compact factors . (Le., if H c Gp is a proper normal CQp-subgroup, then (Gp)cop/Hcop is not compact We recall the standard fact from Lie theory that for k = IR, non compactness of G11 for G semisimple is equivalent to IR-rank(G) > 0 . That the same result is true for any local field of characteristic 0 is a result of Bruhat and Tits. See [Prasad 3] for a simple proof. ) Let G =
11 (Gp)CQp ' so that G is
peS
a locally compact group with the product topology. Lattices and actions of such groups arise naturally via "arithmetic" construc tions. For example, let us assume oo E S. Let Zs be the subring of CQ consisting of all rational numbers whose denominators have all prime factors lying in S. Clearly, if S # { oo }, Zs is dense in IR. However, we have a natural map Zs --+ CQ P for each p E S, and hence a natural map Zs --+ 11 COP · Th� image of Zs will then peS be discrete and in fact one can show that the image of Zs is a (cocompact) lattice in
11 CQ · More generally, suppose G is a connected algebraic CQ-group. peS P
We then have a natural embedding of Gz, --+ 11 Gcop ' whose image is discrete. peS In fact, we have:
10.1.1
Theorem
[Borel 6]: If G is a connected semisimple CQ-group, then Gzs is
a lattice in 11 Gcop peS This is the prototype of an "S-arithmetic" group. (See Definition 1 0. 1 . 1 1 .)
188
Ergodic theory and semisimple groups
Now let G be as in the first paragraph of this section. Any lattice in G is Zariski dense when projec;ted to each (Gp)rpp (This follows from Theorem 3..2 5 and the remarks following its proof.) A lattice r c G is called irreducible ifTH is dense in G for all H of the form (L)Q)p where L c Gv is an infinite normal proper algebraic (Qv-subgroup. Similarly, an ergodic measure class preserving action of G is called irreducible if the restriction to each such subgroup H is ergodic. Many of the results of the earlier chapters extend to this framework In fact, the proofs we have given generalize in large measure as well once one has developed some further algebraic background concerning the structure of algebraic groups over local fields. We shall therefore first indicate (without proof) one such basic algebraic result With this in hand, the extension of our earlier proofs to our current more general framework will then follow Suppose G is a connected semisimple k-group where k is a local field of characteristic 0. We define G + to be the subgroup of Gk generated by { Uk I U is the unipotent radical of some parabolic k-subgroup of G } . Then G + c Gk is a normal subgroup. (See [Tits 1 ] and [Bore!-Tits 2] for a full discussion of G + ) If Gk is compact (ie., k rank ( G ) = 0), then G + = {e}. On the other hand, for k = IR, and G almost IR-simple, if Gn:< is not compact then G + = G2 An analogous result in the case of an arbitrary k is given by the following theorem of Platonov verifying a conjecture of Kneser and Tits.. ..
-
10.1.2 Theorem [Platonov 1] (Kneser-Tits conjecture): Suppose k is a local field of characteristic 0 and G is a connected (algebraically) simply connected, almost k-simple k-group with k-rank (G) > 0.. Then G + = Gk If j:G --> H is a k-isogeny, then / ( G +) the following consequence.
=
H + Hence, Platonov's theorem has
1 0. 1.3 Corollary: Suppose k is a local field of characteristic 0 and G is a connected almost k-simple k-group with k-rank (G) > 0. Then [Gk:G + ] < oo.
The proof o f the vanishing o f matrix coefficients for unitary representations of connected simple Lie groups (Theorem 2 ..2..20), then yields the following result. 1 0. 1.4 Theorem [Howe-Moore 1] (Cj 2.2..20): With our assumptions as in 10 .. 1 .3, suppose n is a unitary representation of Gk so that niG + has no invariant vectors. Then the matrix coefficients of n vanish at oo in Gk .
This result enables us to establish analogues of all the versions of Moore's ergodicity theorem that appear in Chapter 2 .
1 89
Generalizations to p-adic groups and S-arithmetic groups
The following generalized versions of the stiperrigidity theorems (5. 1 .2, 5.. 2 .. 5 ) then follow with essentially the same proof With G = n (Gp)rop as above, we
peS
let rank (G)
=
L �p rank (Gp ) s
-
1 0.1 .5 Theorem [M argulis 1 ] : Suppose each GP is (algebraically) simply connected, (Gp)rop has no compact factors, and rank (G) � 2. Let r c G be an irreducible lattice. Let k be IR, IC, or �' for r prime and H a connected k-simple k-group. Suppose n:r --> Hk is a homomorphism with n (r) Zariski dense in H Then either (a) n(r) is compact (where the closure is in the H ausdorff topology); (b) For some p E S - { oo }, k = �p, and there exists a k--rational surjection qJ :Gp --> H such that the following diagram commutes. r
�r Hk
or c) k = IR or IC and there exists a k-rational surjection G oo --> H such that the corresponding diagram as in (b) commutes. 10.1.6 Theorem (Superrigidity for cocycles): Let G be as in Theorem 10. L5 and suppose X is an irreducible G-space with finite invariant measure. Let k, H be as in 1 0. 1 S. Suppose a:X x G --> H k is a cocycle that is not equivalent to a cocycle into a proper subgroup of Hk of the form Lk where L c H is a k-subgroup. Then either (a) a is equivalent to a cocycle taking values in a compact subgroup of Hk; (b) For some p E S - { oo }, k = �p, and there exist a k-rational surjection qJ:Gp -+ H and a cocycle fJ � a such that the following diagram commutes
or (c) k = IR or IC and there exist a k-rational surjection and a cocycle fJ � a such that the corresponding diagram as in (b) commutes. As in Chapter 5, from these results we deduce the following rigidity theorems.
190
Ergodic theory and semisimple groups
Theorem [Margulis 1 ] [Prasad 2] ( Cf 5. L L): Let S, S' be finite subsets of For p E S, let Gp be a connected, adjoint, semisimple {Qp group so that ( Gp )
10.1.7 V
G'
=
f1 (G�)rop' and assume rank (G) � 2. Let r G, ['
pES'
c
c
pES
G' be lattices, with
r
irreducible. If n:[ � r' is an isomorphism, then S = S' and for each p E S there is a {Qp-regular isomorphism GP � G� such that the product map G � G' is an extension of n Proof: Let Gp be the algebraic universal covering of Gp and G = IT (Gp)
have a natural map q : G � G whose image has finite index in G. (The map is not a priori surjective since we have passed to the k-points for the various fields k in question. ) We can choose an irreducible lattice A c G such that q (A) c r is a normal subgroup of finite index Then n(q (A)) is normal and of finite index in r' and hence is also a lattice. In particular, the projection of n (q (A) ) onto each factor L of G' is Zariski dense. We can therefore apply super rigidity ( 1 0. 1 . 5) to the maps A � Lrop given by composing n o q with the projection onto Lrop · For each p, we thus obtain a (Qp-rational map (/)p:Gp � G � such that the associated product map (/) = IT(/)p:G � G' satisfies $ l A = n o q .. Since G � is an adjoint group, (/)P factors to a map cp p : GP � G� and hence (/) factors to a map cp:G � G' such that c/Jiq (A) = n . By Zariski density of the image of n (q(A) ) in (G�)rop' cp p ( G p) is Zariski dense in G� and hence cpp is surjective . Since q (A) is an irreducible lattice in G and cp (q (A)) is discrete, dim (ker cpp) = 0, and since Gp is an adjoint group, cpp is injective. Thus, cpp is an isomorphism defined over (Qp, and hence cp is an isomorphism To prove the theorem it suffices to show that cfJir = n. If y E r and A. E q (A), we have cjJ (yA.y - 1 ) = n (yA.y - 1) since q(A) c r is normal and c/Jiq(A) = n. Therefore [n(y) - 1 cp(y)] cp(A.)[cp(y) - 1 n(y)] = n(A.) = cp(A.), i e , n(y)- 1 cp(y) commutes with cp(q(A)). Since the projection of cp(q(A)) onto each GP is Zariski dense, the projection of n(y) - 1 cp(y) commutes with all GP However, Gp is an adjoint group for all p, so we deduce that n(y) - 1 cp (y) = e, completing the proof. As with Theorem 5. 1. 1, Theorem 1 0.. 1. 7 is true under weaker hypotheses than rank (G) � 2. See [Prasad 2] for full details. Using 1 0.1 . 6 in a similar manner one can prove: 10.1.8 Theorem (Cf. 5.2.1): Let S, S', G, G' be as in 1 01.7 Let (X, Jl.) (resp.. (X', Jl.')) be an irreducible G-space (G'-space) which is essentially free and with finite invariant measure. Ifthe G-action on X and the G' action on X' are orbit equivalent, then G � G' and modulo this isomorphism, the actions are conjugate.
191
Generalizations to p-adic groups and S-arithmetic groups
10.1.9 Corollary ( Cj 5 ..2.2): Suppose p, q E V Let G(G') be an almost COp-simple (almost COq-simple) group, and r c G
Arguing as in section 9.4, we can also deduce the following property of lattices in p-adic groups acting on manifolds 10.1.10 Theorem (Cf 9.4.1 5): Let G be an almost simple COp-group where p E Z is a prime. Suppose r c G
We remark that by a result of [Borel-Harder 1] every group of the form Gk where G is a semisimple k-group (k a local field of characteristic 0) admits a lattice subgroup. Margulis' s arithmeticity theorem (6.. 12) can be generalized to the present fl:amework as welL We make the following definition. 10.1. 1 1 Definition: Let S be a finite subset of V, and assume oo E S.. For each p E s let Gp be a connected, adjoint, semisimple cop-group for which every simple
factor has COp-rank > 0. Let G
=
0 (Gp)
We call r an S-arithmetic lattice if there exist: (1) a semisimple algebraic CO-group H; (2) for each p E S a decomposition H = H 1 ,p x H z.p where H;,p are semisimple cop-groups; (3) a compact open subgroup K c 0 (H z.p)
peS
and (4) an isomorphism of topological groups 0 (H l ,p)
peS
such that for H *
0 (H 1 ,p)
10.1.12 Theorem [Margulis 1 ] (S-arithmeticity theorem): Let G be as in 10J J 1 and let r c G be an irreducible lattice. Assume rank (G) � 2. Then r is S-arithmetic.
! 92
Ergodic theory and semisimple groups
Proof: The proof is similar to that of the arithmeticity theorem (6. 1 .2) using 10. 1..5 instead of 51 ..2. When convenient, we shall therefore refer the reader to the proof of 6. L2. By passing to a subgroup of r of finite index we can assume
r = q (A) for some irreducible lattice A c G = IT (Gp)
, Poo I ---�
proj
H ro---------;:,. H
Since 1X (r oo) is Zariski dense in H, we clearly have c/JL(r) is Zariski dense in L By Theorem 10. 1 ..5 we deduce that: (i) if a if S, cPL(r) has compact closure; (ii) if a E S, either c/JL(r) has compact closure or c/JL extends to a homo morphism c/JL:G --+ L of the form cPL = 1/JL o Pa where Pa:G � (Ga)roa is projection and 1/Jr:Ga -+ L is a �a--rational surjection. By passing to a subgroup of finite index in r, we can assume that in case (ii), if cjJL(r) has compact closure, then it is contained in a compact open subgroup. (For a = oo, this follows from the fact that c/JL{r) is Zariski dense and hence Ln:< itself would be compact For a =1= oo, this follows from the fact that Lroa admits compact open subgroups.) Combining the maps cPL in (ii), we deduce that for a E S we can write H = H l.a x H z,a as a product of �a-groups, find a compact open subgroup Ka c (Hz,a)roa• and a rational �a-map c/Ja:Ga ·-+ (H l,a) such that, setting K = IIKa,
x K c [I Hroa • and cjJ = ITc/Ja:G � fl (H l ,a)ro" ' we have a a a (1X 0poo)(r) c H * and (1X o p oo ) = (c/J, A.) where A:r � K Each map c/Ja:Ga � (H l,a) is surjective by the argument in the next to last paragraph of the proof of Theorem 61 .2, using now the Zariski density of 1X (r oo) in Hroa· F urthermore, using case (i) above, the argument of the final paragraph of the proof of 6. 1 .2
H* =
[I (H l ,a)roa
193
Generalizations to p-adic groups and S-arithmetic groups
shows that 1X (r co ) n Hz, is of finite index in 1X(r eo ). In particular, (1X a p 00 ) (r) is discrete. Since r is irreducible, and G is a product of adjoint groups, cjJ must be injective. Thus cjJ is an isomorphism. Applying the map r'
H * --� Il (H l .a )Q!a --� G a
to the finite extension ( (IX o PcoHr) n Hz,) c: (1X o Pco Hr), the argument of the final sentences of the proof of Theorem 6.. 12 completes the proof of Theorem 10.. 1 . 1 2. In conclusion, we remark that the arguments of Chapters 7 and 8 apply in the present more general context as welL Thus, we have: 1 0. 1. 1 3
Theorem
[Margulis 6] [Kazhdan 1 ] (Cj 8. 1.2): Let S be a finite subset of
V andfor each pES, let Gp be a connected semisimple ([;p-group. Let G
Il (Gp)Q!P peS and let r c: G be an irreducible lattice. Assume rank( G) � 2. If N c: r is a normal subgroup, then either N c: Z(G) (the center of G) and is finite, or rjN is finite. =
Appendices A.
Borel spaces
In this appendix we collect (largely without proof) some of the basic general facts about Borel spaces. An excellent presentation of much of this and related material is in [Arveson 1]. A Borel space is a set X together with a O"-algebra f!J of subsets of X, called the Borel sets of X. We also refer to f!J as a Borel structure on X. If d is any family of subsets of a set X, there is a unique smallest O"-algebra containing d which we call the Borel structure generated by d. If X is a topological space, then X is a Borel space in a natural way, the Borel structure being that generated by the open (equivalently, closed) sets. If X is a Borel space and A c X is any subset, then A is a Borel space with the Borel structure {A n BIB c X Borel} . If X, Y are Borel spaces, then f:X --+ Y is called Borel if A c Y Borel implies I 1 (A) c X is Borel. X and Y are called isomorphic Borel spaces if there is a bijective Borel map f:X --+ Y with f - 1 Borel A Borel space X is called countably separated if there is a countable family of Borel sets {A;} which separates points. X is called countably generated if there is a countable family {A;} which separates points and generates the Borel structure. Clearly any subset of a countably separated (generated) Borel space is countably separated (generated). -
A.l Proposition: (i) X is countably separated if and only if there exists an injective Borel map X --+ [0, 1 ] . (ii) X is countably generated if and only if X is isomorphic to a subset of [0, 1 ] .
Proof: let { A;} be a countable family o f Borel sets which separates points. Let (/)
Q = fl {0, 1 } Define f:X --+ Q by I (a); = X A, (a) where XA is the characteristic 1 function of A . Then f is an injective Bore] map and f (A;) is Borel in f (X). As is well-known, Q is homeomorphic to the Cantor set in [0, 1], and from these remarks the result follows. A.2 Definition: A Borel space is called standard if it is isomorphic to a Borel subset of a complete separable metric space.
195
Appendices
(We remark that in (ii) of A. l, X need not be isomorphic to a Borel subset of [0, 1]. To the author's knowledge, there is no purely "Bore! theoretic" charac terization of standard Bore! spaces . ) Standard Bore! spaces of course include many spaces arising naturally in geometry and analysis, and they have a number of very strong regularity properties.
Theorem: Any standard Bore! space is either finite, isomorphic to Z, or isomorphic to [0, 1 ]
A.3
Theorem: Suppose f:X --+ Y is an injective Bore! map where X is standard and Y is countably separated.. Then f (X) c Y is a Bore! set andf:X --+f (X) is an isomorphism. (In particular, f (X) is standard).
A.4
If / is not injective in A4, f(X) may not be Bore! However, the following presents another general situation when it will be Bore!..
[Kallman 1]: Suppose X is a standard Bore! space and Y is a separable topological space metrizable by a complete metric . Suppose A c X x Y is Bore! and that for each x E X, Ax = { y E Yj(x, y) E A} is a countable union of compact sets. Let f: X x Y -+ X be projection. Then f (A) is Bore! and there is a Bore! section f (A) --+ A of the map f lA . A.5
Theorem
A.6 Corollary: Suppose X, Y are metrizable by complete separable metrics. Let f:X --+ Y be a Bore! map such that for each y E Y, f - 1 ( y) is a countable union of compact sets . Then f (X) is Bore! and there is a Bore! section f (X) -+ X off. Proof: Apply A.5 to A = { (x, y) E X onto Y From A. 5 one can also deduce:
x
Yjf(x) = y} and the projection of X
x
Y
A.7 Theorem [Kallman 1 ] : If X is a standard Bore! G-space where G is a locally compact second countable group, and the action of G on X is smooth (i.e. X/G is countably separated), then X/G is standard and there is a Bore! section X/G -+ X of the natural projection.
! 96
Ergodic theory and semisimple groups
If H c G is a closed subgroup of a locally compact (second countable) group, then there is a Bore! section G/H - G.
A.8
Corollary:
Remarks: A.5, A 7 are not in [Arveson lJ See [Kallman 1] for the proof. A.8 was first observed by Mackey.. If G is a Lie group then A.8 is a straightforward consequence of the implicit function theorem In A 6, if X and Y are compact and / is continuous, the existence of a Borel section was first due to Federer and Morse. (See [Parthasarathy 1 ] for a proof in this case . )
Although the image of a standard space under a Borel map need not be standard in general, in the presence of a measure it is "almost standard. " As usual, if X is a Borel space and J.l is a measure defined on the Borel structure of X, a set A c X is 11-null if there is a Borel set B with J.l(B) = 0 and B ::::J A, and 11-conull if its complement is ,u-null..
Let X be a standard Bore! space, Y countably generated and f:X - Y a surjective Bore/function. Suppose J.l is a probability measure on Y.. Then there is a conull Bore! set Z c Y such that Z is standard, and there is a Bore! section Z - X off A.9
Theorem:
The existence of a section defined on a conull set as in A.9 is the von Neumann selection theorem. If X is a compact metric space the presence of a measure also clarifies the relation of a Borel set to the topology.
is a compact metric space with a probability measure. a Bore! set, then there is a Bore! set B c A which is conull in A and If A c X is is a countable union of compact sets. A.lO
Theorem: Suppose X
For a proof, see [Parthasarathy 1]. The analgoue of Theorem A.3 in the presence of a measure is also true. Theorem: Let X be a standard Bore! space and 11 a probability measure on X with no atoms. Then there is an isomorphism fX - [0, 1 ] such that f* (J.l) is Lebesgue measure. A.l l
We recall that if X is a standard Borel space with a probability measure, then B(X) denotes the measure algebra of X. We can identify B (X ) as a closed subset of Loo (X) with the weak-*-topology. H f:X - Y is a measure class preserving Borel map, there is an induced map f * :B(Y) -+ B(X ) which is continuous, Boolean operation preserving, and injective.
197
Appendices
Theorem: Suppose X, Y are standard Baret spaces with probability measures and F:B( Y) -+ B(X) is a continuous, Boolean operation preserving, injective map. Then there is a measure class preserving Bore/ map f: X -+ Y such that f * = F A.12
Given a continuous family of such maps F, the point maps can be chosen to vary in a Bore! fashion .
Suppose Z is a space metrizable by a complete separable metric and that for each z E Z, Fz:B( Y) -+ B (X) is a continuous, Boo lean operation pre serving, injective map. Assume also that the map B ( Y) x Z -+ B(X), (A, z) -+ Fz(A) is continuous. Then there is a Bore/ map f:X x Z -+ Y such that for each z E Z, /z (x) = f (x, z) is a measure class preserving Bore/ map withf: = Fz. A.13
Theorem:
For a proof of A 1 2, A 1 3, see [Ramsay 1 ] (Theorems 2.1, 3. 3). B.
Almost everywhere identities on groups
In this appendix we collect some facts concerning almost everywhere identities related to locally compact groups . Throughout, G will denote a locally compact second countable group with Haar measure m. B.l Proposition: H = G.
Let H c G be a conull subset closed under multiplication.. Then
Proof: Since H is conull, so is H - 1 Let x E G . Then H - 1 x is conull and hence so is H - 1 x n H. Thus, in particular, there exists h E H such that h - 1 x E H. Hence X E hH e H.. .
Suppose J is a topological group whose Bore/ structure is coun tably generated (e.g. J second countable).. Let n:G -+ J be a Bore/ function such that for almost all (x, y) E G x G, n (xy) = n (x) n (y). Then there is a Bore/ homo morphism n0:G -+ J such that n = no a. e. B.2
Theorem:
Proof: As a Bore! space we can assume J
c
[0, 1] by A. L Let
H = { y E G ix -+ n (x) - 1 n (x y) is an essentially constant function on G }.
By Fubini, H is conu!L Further, if y, z E H, writing
1 98
Ergodic theory and semisimple groups
it is clear that yz E H, and hence by BJ that H = G. The same equation shows that if n 0 (y) E J is such that n (x) - 1 n (xy) = n0 (y) for a. e . x, then no:G -+1 is a homomorphism Finally no is Bore! by Fubini, since (remembering J c [0, 1] ) no ( y) = J n (x) - 1 n (xy) dm' (x) where m' is a probability measure in the class of Haar measure. .
B.3 Theorem (Mackey): Suppose J is a second countable topological group and n:G � J is a measurable homomorphism. Then n is continuous.
n is surjective. Let U c J be an open neighborhood of the identity and V c U a symmetric open neighborhood with V2 c U. Let { Yn } be a countable dense set in J and n (g.) = Yn · We have H = u V· y. and hence Proof: We can assume
G = U n - 1 ( V) g •.. It follows that for some n, m (n - 1 ( V) g.) > 0 so m(n - 1 ( V) ) > 0.
n
Since n - 1 (U) :::> n - 1 ( V) n - 1 ( V), it suffices to show:
G is a compact set in G with m (A) > 0, then A - l A contains a neighborhood of e E G.
B.4 Lemma: If A
c
Proof: If Axn A -1=- 0, then x E A - 1A. So it suffices to show that {xE GJAxnA -1=- 0} contains a neighborhood of e. Since A is a compact set of positive measure, we can find an open set W => A such that m ( W) < 2m(A).. Again by compactness of A, we can find a symmetric neighborhood N of e such that Ax c W for all x E N. Since m (Ax) = m(A) and m ( W') < 2m(A), we must clearly have Ax n A #- 0 for
X E N. In 2.2. 1 6, we showed that an essentially invariant function on a G-space agrees a.e. with an invariant function. We now consider the analogous fact for maps between G-spaces.
Proposition: Suppose (X, J.l) is a standard Borel G-space with quasi-invariant measure and that Y is a standard Borel G-space. Supposef:X � Y is Borel and for each g EG,f (xg) = f(x)gfor almost all x. Then there is a G-invariant conull Borel subset X 0 c X and a Borel G-map rx 0 � y such that I = l a. e. B.5
Appendices
1 99
Let X0 = {x E X i g l-4f(xg) g - 1 is essentially constant in G } . For x E Xo, let J(x) E Y be such that f (xg)g - 1 = l (x) for a. e.. g E G. As a Borel space we can assume G c [0, 1] by A l, and hence by Fubini Xo is conull and ! is Borel (by an argument as in B. 2) . We have f = Ja e , and from the expressionf (xhg) g - 1 = [f (xhg) (hg) - 1 ] h, we deduce that for x E X0, h E G, we have xh E Xo and lcxh) = J(x) h Proof:
Suppose now that (X, fJ.), ( Y, v) are standard G-spaces with quasi-invariant measure . Let B (X, fJ.), B( Y, v) be the corresponding measure algebras. (Cf. 8.1). B.6 Corollary: Suppose cp:B( Y, v) --+ B(X, fJ.) is a continuous infective Boo lean homomorphism which is a G-map.. Then there is a conull G.-invariant Bore! set X o c X and a G-map J:x o --+ Y such that 1 * = cp. Proof: By general measure theory there is a Borel mapf:X --+ Since cp is a G-map, f will satisfy the hypotheses of B. 5.
Y such that [ * = cp.
B.7 Corollary: Suppose (X, fJ.), ( Y, v) as in B.6 and that cp is a bijection.. Then X and Y are conjugate G-spaces . le , there are conull, Bore!, G.-invariant sets X0 c X, Y0 c Y and a measure class preserving Bore! isomorphism } X 0 --+ Yo which is a G-map. Proof: Choose!as in B..6 . Since cp is surjective from general measure theory there is a conull Borel set X 1 c X o on which ! is injective . Let X 2 = {x E X 1 lxg EX 1 for almost all g E G } . By Fubini 's theorem, X2 is conull and Borel. We claim that lis injective on X2 G. Suppose x, y E X2 and J(xa) = Tc yb) for a, b EG . Since x, y E X 2, there is some g E G such that xag, ybg E X 1 . (In fact, this is true for almost all g ..) Since l is a G-map, we have J(x ag) = Tc ybg), and by injectivity ofl on X 1 , xag = ybg. Thus, xa = yb, establishing injectivity on X 2 G. By A.4, it clearly suffices to see that X 2 G contains a conull, G-invariant Borel set.. This follows from the generally useful:
Suppose X is a standard Bore! G-space with a quasi-invariant measure, and that A c X is Bore!.. Then there is a Borel subset B c A which is conull in A, and such that. (i) BG is Bore!; B.8
Lemma:
200
Ergodic theory and semisimple groups
(ii) There is a Borel map cp:BG -+ G such that for all x E BG, xcp (x) E B, and for all x E B, cp (x) = e. Proof: By 2. 1.1 9 and AA, we can assume that
X is compact metric and that G acts continuously on X. By AJO there is a Bore! set B c A which is conull in A and is a countable union of compact sets B; .. Since G is also a countable union of compact sets, BG will be as well, and in particular BG is BoreL To see (ii), it suffices to construct for each compact K c G and each i, a Borel map cp:B;K -+ K such that x · cp (x) E B; for all x E B;K. However, this follows from A.5 We now turn to the relationship between cocycles and strict cocycles, notions introduced in section 4.2. We recall that for transitive G-spaces, this relationship is clarified by 4..2J 5. Here we deal with the general case. Theorem: Let S be a standard Borel G-space with quasi-invariant measure Suppose H is a topological group whose Bore[ structure is countably generated (e.g. H second countable). Let et.:S x G -+ H be a cocycle. Then there is a strict cocycle [J:S x G -+ H such that for all g E G, [J(s, g) = et. (s, g) for almost all s E S. B.9
f.L
X = { (s, g) E S x G[h -+ et. (s, gh) et. (sg, h) - 1 is an essentially constant H-valued function of h E G } . For (s, g) E X, let & (s, g) be this constant. Then Fubini's theorem (recall H c [0, 1 ] as a Bore! space by AJ) shows that X is a conull Bore! set, &:X -+ H is Bore! and a = et. a. e. Suppose (s, g), (sg, a) E X. Then Proof: Let
the relation
et. (s, gah) et. (sga, h) - 1 = [ et. (s, gah) et. (sg, a h) - 1 J [et. (sg, ah) et. (sga, h) - 1 J implies that (s, ga) E X and a(s, g) &(sg, a) = &(s, g a). Now let So = {s E S[ for almost all g s G, (s, g) E X and (sg, g - 1 ) E X} . Since X is conull and Bore! in S x G and (s, g) -+ (sg, g - 1 ) is measure class preserving, it follows from Fubini's theorem that So is a conull Bore! subset of S. We now claim that if s E So and a E G with sa E S0 , then (s, a) E X. By the preceding paragraph it suffices to show that for some g E G, we have (s, ag), (sag, g - 1 ) E X. Since s E So, { g[(s, ag) E X} is conull in G, and since sa E So, { g[(sag, g - 1 ) E X} is conull in G. Therefore, our assertion follows. Summarizing we have produced a conull Borel set So c S such that if s, sg, sgh E So for g, h E G, we have &(s, gh) = a(s, g)&(sg, :h). By passing to a conull Bore! subset, we can also assume So satisfies the conditions Of B.8. Thus, S0 G is a conull Bore! G-invariant set and we can find a Bore! map .
201
Appendices
cp:S 0G -+ G such that cp(s) = e for sE So and s· cp(s)E S0 for all sES 0 G. We now define [3:S 0 G x G -+ H by [J(s, g) = & (scp (s), cp (s)- 1 gcp (sg)). From the conclusion of the preceding paragraph, it follows that [3 is a strict cocycle. Finally, for s t/= S0 G, define [J(s, g) = e for all g. Then [J:S x G -+ H is a strict cocycle and [3 = a a.e To complete the proof, let Go = { g E Gia(s, g) = [J(s, g) for almost all s} .. By Fubini, Go c G is conull and from the cocycle identity we deduce that G 0 is closed under multiplication. From B.l, it follows that Go = G. We now show that a continuous G action on B (X) actually defines a G action on points.
B.l O Theorem [Mackey 2]: Let (X, J1) be a standard Bore! space with a prob ability measure. Suppose G acts continuously on B(X) so as to preserve the Boolean operations. Then there is a standard Bore! G-space Y with quasi-invariant measure v, conull Bore! sets X 0 c X, Y0 c Y, and a measure class preserving Bore/ iso morphism cp:X 0 -+ Yo such that cp * :B( Y) -+ B (X) is a homeomorphic G-map.
a:X
x
G -+ X such that for each g E G the map ag:X -+ X given by ag(x) = a(x, g) is a measure class preserving map with a; equal to the original action of g - 1 on B (X). It follows that a is almost an Proof: By A. 1 2, there is a Borel map
action. I.e., (i) for g, h E G, a (x, g h) = a(a(x, g), h) for almost all x E X; (ii) for each g E G, a(a(x, g), g - 1 ) = x a.e. Choose an injective Borel function f:X -+ [0, 1]. Define h:X -+ Loo (Gh (the latter being the unit ball in L 00 (G), with the weak- * -topology) by (h(x))(g) = f (a(x, g)).. Let X 0 = {x E XIa(a(x, g), g - 1 ) = x for almost all g E G } . By (ii) and Fubini, Xo is a conull Borel subset of X. If x, y E X and h (x) = h (y), then for almost all g E G, a(x, g) = a(y, g).. Thus if x, y E X 0 , we deduce x = y, i.e.. h is injective on X 0 Then one checks that ( Y, v) = (L00 (G) l , h * (J1)) and Y0 = h (Xo) satisry the required conditions . (For a proof verifying all details, see [Ramsay 1 ] . )
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Subject Index adjoint group, 37 affine action, 60 affine space over S, 78 affine variety, 32 algebraic hull of a cocycle, 167 algebraically simply connected, 37 algebraic universal covering, .37 almost invariant vectors, 1 30 almost k-simple, .37 ex-invariant, 72 ex-twisted action, 65 amenable action, 78 amenable group, 59 arithmetic subgroup, 1 14 barycenter, 6 1 Bernoulli shift, 1 77 Boolean a-algebra, 1 50 Borel density theorem, 4 1 Borel map, 1 94 Borel space, 1 94
entropy, 1 76, 1 77 equivalent cocycles, 65 ergodic, 8 essentially constant, 1 1 essentially free, 68 essentially invariant, 2 1 essentially rational, 5 5 essentially transitive, 8 Gaussian, 1 10 induced action, 75 induced representation, 74 irreducible action, 20 irreducible lattice, 1 8 irreducible variety, .32 isogeny, .37 isomorphic actions, 8 Kakutani-Markov theorem, 59 k-almost algebraic, 40 Kazhdan property, 1 .30, 1 65 k-cocompact, 47 k-group, .3.3 Kneser-Tits conjecture, 1 88 k-rank, 8 5 k-simple, .37 k-split, 85 k-variety, 33
cocompact, 1 cocycle, 65 cocycle reduction lemma, 1 08 Comm (1), 1 22 commensurability subgroup, 1 22 commensurable, 3 connected algebraic group, 35 contracting automorphism, 1 52 convergence in measure, 49 countably generated, 1 0, 1 94 countably separated, 1 0, 1 94
lattice, 1 Lebesgue density theorem, 1 54 local field, .34
derivative cocycle, 67 direct integral, 23 Dye's theorem, 82
Mackey range, 77 matrix coefficient, 2.3 mean, 1 .3 .3
Su�ject index
measure algebra, 1 50, 1 96 measure class, 8 minimal action, 1 83 modular cocycle, 83 modular flow, 83 Moore's ergodicity theorem, 1 9, 21 multiplicative ergodic theorem, 1 78 orbit equivalence, 68 parabolic subgroup, 47 P(G), 1 3 3 Poincare recurrence, 1 65 properly ergodic, 8 property T, 1 30 quasi-invariant, 8 quasi-projective variety, 33 Radon-Nikodym cocycle, 67 rank, 85, 1 8 8 rational function, 32, 33 regular function, 32, 33 restriction of scalars, 1 1 6 IR-rank, 8 5
209
S-arithmetic, 1 9 1 skew product, 7 5 smooth, 1 0 stabilizer, 1 2 standard Borel space, 1 94 strict cocycle, 65 strictly equivalent, 65 superrigidity, 85, 98, 1 89 submatrix, 143 tempered cocycle, 178 twisted action, 65 UCB(G), 1 36 unipotent group, 52 unipotent representation, 53 variety, 33 von Neumann selection theorem, 196 weak containment, 143 Zariski topology, 32
