4-dimensional compact projective planes with a 7-dimensional collineation group

DIETER BETTEN

4-DIMENSIONAL

COMPACT

WITH A 7-DIMENSIONAL

PROJECTIVE

PLANES

COLLINEATION

GROUP

Dedicated to H. R. Salzmann on his 60th birthday

ABSTRACT.The classification of 4-dimensional compact projective planes having a 7-dimensional collineation group is completed. Besides one single shift plane all such planes are either translation planes or dual translation planes.

1. INTRODUCTION In [28], [29] Salzmann studied 4-dimensional compact projective planes; that is, topological planes homeomorphic to the projective plane over C. For the group E of continuous collineations he proved: Z is a Lie group of dimension ~ 16, and if dim E ~> 9, then the plane is arguesian, i.e. isomorphic to the plane over C. For dim Z = 8, the plane is isomorphic or dual to a translation plane [30]. This result initiated the study of 4-dimensional translation planes [1], [25], and all such planes with dim Z = 8 and also with dim E = 7 could be classified, see [5] and papers cited there. A topological projective plane is called flexible if the collineation group has an open orbit in the space of flags (flag = incident point-line pair). A 4-dimensional translation plane is flexible if and only if dim E~> 7, therefore the above result can be restated in the form: all flexible 4-dimensional translation planes are known. If one wishes to classify all 4-dimensional compact flexible projective planes, one can now assume that the plane is neither isomorphic nor dual to a translation plane. Since flexibility implies dim E >~ 6, only the dimensions dim E = 7 and dim Z = 6 have to be considered. One class of such planes are the so-called shift planes, which can be generated by 'shifting' the graph of a suitable function f : ~2_~ ~2. The 4-dimensional shift planes were first considered in [4], and all flexible shift planes are now classified [13], [15], [7]. Among all shift planes there is exactly one which has a 7-dimensional collineation group (Knarr 1983 [13]). This plane is generated by the function f: ~2 _._+~2: (X, y)~-~ (xy -- x3/3, y2/2 - x4/12); it will be called the shift plane of Knarr in the sequel. Another class of planes, which are neither isomorphic nor dual to translation planes are planes possessing a 3-dimensional translation group Geometriae Dedicata 36:151 - 170, 1990. © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

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with respect to some fixed axis [6], [35]. But, whereas the classification of 4dimensional flexible planes has made good progress, up to now the situation dim 2 = 7 has not been clarified. The purpose of this note now is the proof of T H E O R E M . Every 4-dimensional compact projective plane with a 7-dimensional collineation group is either isomorphic/dual to a translation plane or isomorphic to the shift plane of Knarr.

2. PROOF OF THE THEOREM In order to simplify notation we denote by E the connected component of the identity of the group of all continuous collineations. Let P and ~ be the space of points and the space of lines of the plane. (1) The group Z is solvable and fixes a line Wand a point v ~ W. We may assume that no further line or point is fixed by Y,. Proof In [19] R. L6wen studies 4-dimensional compact projective planes where E is nonsolvable and has dimension at least 6. He proves that either E is isomorphic to the 6-dimensional group R 2" GL~-~ or the plane is a translation plane or the dual of a translation plane. From this result it follows that E is solvable. In [5] it is proved that E fixes a flag; it is also shown there that a further fixed element implies that the plane is a translation plane. (2) We may assume that ~. has no subgroup isomorphic to ~4. Proof. In [16] Knarr proves that a subgroup isomorphic to R 4 can occur only in the cases: shift planes, translation planes and duals of translation planes. There is exactly one shift plane with 7-dimensional collineation group, generated by the graph of the function (x,y)~-~(xy-xa/3, y2/2 -- x4/12), [13, Satz 4.8]. (3) Up to duality, Y~acts transitively on ~v\{W}. This action will be called the horizontal action in what follows. Since ~dv\{ W} is homeomorphic to R 2, we can introduce horizontal coordinates ~v\{W} = {(x, y) lx, y e ~ } . We denote by Stx,r) the line of ~v\{W} having horizontal coordinates (x, y). Proof We show that Z is transitive on ~v\{W} or transitive on W\{v} and dualize, if necessary. Assume that both actions were not transitive. Since by (1) we have no further fixed point or fixed line, this would imply a 1dimensional orbit on ~ \ { W } and a 1-dimensional orbit on W\{v}. Fixing two lines Y, Y' and two points u, u' from these orbits would give a 3dimensional group. Fixing another point 0 ~ Y would give a quadrangle (four points, no three of which are collinear), fixed by a group of dimension at least

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1. This is a contradiction to the following Lemma on quadranoles [28, 4.1]: If the collineation ~0 is contained in the connected component of the collineation group and fixes the points of a quadrangle, then tp = 1. In the following we use the list of Mostow of all transitive actions of connected Lie groups G on surfaces [18], [23] also described very explicitly in [3]. We observe that for G solvable the list becomes rather short: Besides affine actions (which have ~2 as a regular normal subgroup) there are only the following solvable connected Lie groups transitive on N2 = {(x, y) lx, y ~ R}:

9roup G

dim G

Mostow No.

~n:~ (L 2 x N) ~":,~L 2

n + 3 >~ 5 n + 2 ~> 4

III 7 II 5, III 3

Rn'L2

n

+ 2 >~ 4

III 4

Comment C o m m u t a t o r G' transitive and action on x-coord.: L2

~n>~ L2

n + 2/> 4

III 5

~">,~ L2

n + 1 >/3 2

II3 I 2

C o m m u t a t o r G' not transitive and action on x-coord.:

These non-affine actions have the lines x = const, as sets of imprimitivity. The action on the space of these lines may be an L2-action or an R-action. We need these transformation groups in more detail for dim G = 6 and list the generating vector fields (abbreviation p = O/Ox, q = O/Oy):

p, q, xp, yq, xq, x2q p, q, xp + yq, xq, x2q, x3q p, q, xq, x2q, x3q, xp + ryq (r ~ 1) p, q, xq, x2q, xaq, xp + 4yq + x4q III 5: p, wq, w' q, w" q, w(3)q, yq, where w(x) satisfies the differential equation III 7: II 5: III 3: I I I 4:

W(4)

_~.

C0W(3) -~ Cl W(2) -.~ e2 W, -~- CaW, e i ~

II3: p, wq, w'q, w"q, w(a)q, w(4)q, where w(x) satisfies the differential equation w(5) = Cow(g) + e l W(3) -~- ... -~- e3 Wt dr e4w, e i ~ o these vector fields define the Lie algebra of the acting group, and we get the group by applying the exponential map. We will come across transitive actions which have L 2 as a regular normal subgroup. We note that such actions have dimension at most 4 since dim Aut L z = 2. R E M A R K . The group L 2 ~ Aut 1 L2 is the collineation group (connected

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component) of the real half plane. This transitive action belongs to the Mostow Type III 5. (4) dim E[vj t> 2. Proof Case dim Ztv] = 0. Since by the list of actions given above we can get Etv] by successively fixing elements in simply connected orbits, the group Z[~] is connected and therefore 1. Hence Z/E[~ 1 = X. The action of E on ~,\{W} is 7-dimensional and therefore not affine. F r o m the list it follows that n + {1, 2, 3} = 7 and therefore n >/4 in contradiction to (2). Case dim E[~ = 1. We first show 5". is transitive on P \ IV,, and E[~j ~ E. Otherwise for every line S 9 v, S ~ W, the stabilizer E s would fix some E Horbit in S. Therefore, the 4-dimensional subgroup of E which fixes every line S(o,y), y ~ R, fixes in each S(o,r) some Et~l-orbit. Hence we can construct a contradiction to the lemma on quadrangles. Since Es acts transitively on S\{v}, it also permutes the Etvj-orbits in S. Therefore all orbits under X[,l are homeomorphic to R and also Zt~7 - R. Consequence: The stabilizer Z o, 0 ~ P\W,, is a Lie group homeomorphic to R 3.

To exclude the horizontal action II 3, we note that in this case the 3dimensional stabilizer E o, 0 ~ S(o,o), is commutative and fixes every line S(o,r), y E R. If Eo is transitive on some of these lines S' then the 1-dimensional group Eo, p, p ~ S', fixes all points in S', a contradiction. If E o is not transitive on S', then by fixing points in S(o,o) and in S' one can get a contradiction to the lemma on quadrangles. Next we want to exclude the horizontal action III 5. We show: X o, 0sP\W,, is isomorphic to ~2>,~ {(e)l t ~ e t t ~} and acts transitively on W\{v} in the affine way (group of dilatations). Proof The structure of E o can be deduced by commuting the vector fields in III 5. I f E o is not transitive on W\{v} then Eo has on W\{v} an R-orbit on which it induces an L2-action. The kernel of this action is a 1-dimensional normal subgroup of Zo and therefore contained in R 2. So it fixes every line S~o,r~, y ~ R, and therefore a quadrangle, a contradiction. Weclaim:ZtvI = Ztw I and the action of Z/Ztw I on W\{v} is equivalent to the action of Z/Zt~1 on P.~\{W}, i.e. is also of type III 5. Proof Since Et,,~ ~ ~ acts regularly on every $ 9 v, S ~ W, the common axis must be W: Zt, 1 c Etw j. Since E is transitive on P\W there are no homologies in Ztw ~, and since Z is transitive on W\{v}, all elations in Etw I have centre v: Z~wl ~ ZH. The group Z/Et~l is isomorphic to ~ 4 : ~ L2 as can be seen by commuting the vector fields of III 5. Other groups in the list which are of the form ~*>~ L2 are only the groups II 5 and II 3. But there the complement L2

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acts differently on ~4, therefore these groups have a different isomorphism type. It follows that the action of Z/Etw ~ ~ Z/Err ~ on W\{v} must also be of type III 5. F r o m this we now get a contradiction: The group I I I 5 on W\{v} is imprimitive and induces an E-action on the space of lines of imprimitivity. But since Z contains the dilatation group Z o, the action induced there must be an L2-action. We come to the actions III 7, II 5, III 3 and III 4 and observe that in these cases the vector field q is an ideal of the algebra. Let e be a generator of Ztvj, then (e, q) is an ideal of £(E). We claim: The ideal (e, q) is commutative and N = exP(e, q) is a normal subgroup of Z isomorphic to N2. Proof For 0 e P \ W the subgroup of E fixing the orbit ON acts faithfully on ON (lemma on quadrangles) and admits N as a regular normal subgroup. If N _~ L z this action can have dimension at most 4, a contradiction to dim Z o = 3. Take the quotient space (P\W)/N -~ Na on which Z/N acts as a transitive transformation group. We show that this action has a regular normal subgroup isomorphic to ~2. Proof Let ~0: E --* Aut N be the h o m o m o r p h i s m induced by conjugation. Since Z o defines a 3-dimensional automorphism group of N = R 2, the image of ~o has dimension at least 3. Since Z is solvable, this dimension is exactly 3. Therefore the kernel of q~ has dimension 4; we denote by K its connected component of the identity. Then K acts regularly on P\W. Otherwise the stabilizer Ko, 0 e P\W, has positive dimension and fixes all points of ON, a contradiction to the lemma on quadrangles. Therefore KIN is a regular normal subgroup of Z/N. If KIN ~ L 2 then the transitive action of Z on (P\W)/N -~ ~2 is contained in L2>~ Aut 1 L 2. We now compare the following two actions: (a) Z/U on (P\W)/N = {(x, u)lx, ueE} and (b) Z/Z~vj on ~v\{W} = {(x, y)lx, y e ~ } , which induce the same operation on the x-coordinate since the lines x = const, are lines of imprimitivity for both actions. As we have just shown, the action (a) is contained in the collineation group of the real half plane. Therefore we have an R-action on the x-coordinate. But the action (b) is one of the Mostow types III 7, II 5, III 3 or I I I 4 and induces therefore an L zaction on the x-coordinate. This contradiction implies that K/N must be commutative and since K is homeomorphic to ~4 we get KIN "~ R 2. It follows that K is a central extension of ~2 = exp(e, q) by R 2 and is therefore nilpotent and homeomorphic to ~4. There are three Lie groups of

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this type:

R4 , N i l x ~

and

R3 ~

{e( ° oJiol xpt

1 0

te

,

1

where we denote Nil = R2>~ {(¢ t)lt e N}, [22; 8]. The last of the three groups has a 1-dimensional centre only, a contradiction. If K = Nil x N then, by construction, N2 = exp(e, q) is the centre of K, which is the product of the centre of Nil and the second factor. The centre C of Nil is also the commutator subgroup K' of K and therefore is normal in £. We claim that C = K' coincides with E[v] = exp (e). For otherwise exp ( e ) and C would be two different 1-dimensional normal subgroups of E contained in R2 = exp(e, q). Therefore the 3-dimensional stabilizer No, 0 e P\{W}, could not act faithfully on R 2, a contradiction. Since K acts transitively on P\W, it also induces a transitive action on £v\{W}. The 2-dimensional stabilizer Ks, S e £v\{ W}, is normal in K since it contains the centre C = exp ( e ) = Z[,,] of Nil. Therefore K s fixes all lines through v in contradiction to dim Z[~] = 1. So, also the second possibility K = Nil x ~ is excluded and we get K = R4 in contradiction to (2). (5) dim Eta] ~ 3. We prove this by using the homology and elation method. This strong tool was first used by Salzmann in [28,4.6], who refined it in [31]. A thorough study was made by H/iN in [10]. We cite here part of his theorem [10, 1.1], which will suffice for our purposes: T H E O R E M O N H O M O L O G I E S AND ELATIONS. Let f~ be a closed connected subgroup of the group of all axial collineations with respect to an axis A, and suppose dim f~ ~> 1. I f the set Z \ A of all centres of homologies in f~ contains at least two points, then the group of elations ~[a,a] is connected and Z \ A is an orbit under f~lA,a]- Moreover, Z \ A is a manifold of positive dimension, closed in P \ A and homeomorphic to ~lA,al. EXAMPLE. Suppose E acts transitively on P \ A and E contains for some point x e P \ A a homology ¢ 1 with centre x and axis A. Then the translation group with respect to the axis A is transitive (and the plane is a translation plane with respect to A). We also describe the situation where only elations occur: T H E O R E M O N ELATIONS. Let ~ be a closed connected subgroup of all elations with respect to an axis A and suppose dim ~ / > 1. I f ~ contains elations

4-DIMENSIONAL COMPACT PROJECTIVE PLANES

157

with different centres, then f~ _~ ~", n >1 1, and the map f: Pl(E") ~ A, which maps every one-parameter subgroup of ~" to the corresponding centre on A, is continuous. Proof. Since there are elations with different centres, the group f~ is commutative. By [28, 3.2] the group f~ contains no compact subgroups ~ 1, therefore f~ ~ E". The m a p fl: f~\{ 1} ~ A: ~o~ centre of ~, is continuous [20, (15)], and therefore also the induced m a p f is continuous. E X A M P L E . If, in the situation above, dim f~ = 2, then f~ -~ E2, and the set of all centres on A is a circle. To prove (5), we first assume dim Etvj >~ 5. Then for every line X with X ~ W and v ¢ X, the homology group Z[~,xI is at least 1-dimensional. Dual application of the theorem on homologies and elations implies that the plane is a dual translation plane. Assume now dim Zt, 1 = 4. We dualize and take dimE[w1 = 4, Z transitive on W\{v}. If all collineations of Ztw ] are translations with respect to the axis W, we have a translation plane. If the group Etw,wj is 3-dimensional, then the orbit ofcentres of homologies with axis W is also 3-dimensional. So we get planes studied in [6], [35], which are obtained from two half translation planes, glued together alone some ~3. All planes of this kind are either dual translation planes or they have collineation groups of dimension ~<6. We next suppose dim Z[w,wl = 2. If there are translations in different directions, then Z[w,wj is commutative and isomorphic to R 2. By the theorem on elations (example) the centres of the one-parameter subgroups of R z form a circle on W in contradiction to the transitivity of Z on W\{v}. Therefore all translations in Ztw,w I have direction v. Since the centres of homologies in Y'tw] form an orbit under Z[w,w], the orbit must be a line Y through v, Y # W. This line Y must be fixed under Z in contradiction to (1). The assumption dim Z[w,wI = 1 implies dim Ztwl ~< 3 and for dim Z[w,w] = 0 we would get a unique centre p of homologies and dim~[w] = dimZ[p,Wj ~< 2, a contradiction. The assertions (4) and (5) show that we have to consider only two cases, namely dim ZE~1 = 2 and dim Z[v] = 3.

2.1. Case dimXtv] = 2 (6) dimEtv,~ ] = 2; dim Et~,r ] = 0 for Y~ v, Y ~ W.. Assume by contradiction that the group of elations with centre v has dimension ~<1, then dual application of the theorem on homologies and elations would give a fixed point w e W, w ¢ v in contradiction to (1).

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Therefore the elation group Etv,vj has dimension at least 2, and since Zrv,oI c Et, j and dim Et~1 = 2, we get dim Et~,vI = 2. Since E acts transitively on !3v\{W}, dual application of the theorem on elations shows that there is no one-parameter subgroup of Et,,, ~having an axis Y with Y ~ W and v ~ Y. Therefore all elations in Eta,, j have axis W and Ztv,~j = Et~,wI is the transitive translation group in direction v (which might be non-commutative). N o w look at the 5-dimensional group E r for Y9 v, Y ¢ W. This group acts transitively on Y since it contains the transitive translation group Et,,, 1. We will study the kernel Err I of this action. (7) I f Err ~ has positive dimension, then dim Err I = 1 and Err I consists of homologies with centre w E IV,, w ~ v. Proof Take a one-parameter subgroup of Ztr 1. The elements of this group have a c o m m o n centre which must be different from v by (6). The centre cannot lie on Y \ { v } since W is fixed. Therefore the common centre is a point w ~ W,, w ~ v. This point w is the only possible centre of homologies in Err 1 and is in particular fixed under Ey. For otherwise by the theorem on homologies and elations we would get a 1-dimensional group of shears with axis Y in contradiction to (6). If dim Err I i> 2, then we get a transitive group of homologies Etw, r j, w ~ W, w # v. This would imply that the 'horizontal' action of E/Err I on ~,\{W} = E2 is 2-transitive. But a 5-dimensional 2-transitive connected group on the plane ~a is isomorphic to ~2>~ SL2(E) [32] and is not solvable, a contradiction. (8) Assume dimZtr I = 1. The induced action of Ey on Y is then 2-transitive. For otherwise we could construct a contradiction to the lemma on quadrangles since w e W is fixed and the horizontal action is not 2-transitive. Since dim Zr/Etrj = 4 the 2transitive action on Y is isomorphic to ~2>,aC* [32], the group of similitudes. The normal subgroup ~2 corresponds to the translation group in direction v, so this translation group is commutative. Now let the group E operate on the translation group ~2 by conjugation. This induces a h o m o m o r p h i s m ~o: E / ~ 2 ~ GL(~2). The image of this homomorphism contains C* and cannot have dimension greater than 2 since all groups considered here are solvable. Therefore the kernel of q~ has the form K/[~2, d i m K = 5, and acts transitively on ~2 = ~2v\{W} (lemma on quadrangles). We need the following lemma: L E M M A O N F R E E STABILIZERS. Let G be a 3-dimensional connected solvable Lie group acting transitively on the plane ~2. Then either the stabilizer Gx, x 6 ~ 2 , fixes a further point x ' v ~ x or the commutator group G' acts transitively on ~2.

4-DIMENSIONAL COMPACT PROJECTIVE PLANES

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This can easily be seen from the structure of 3-dimensional Lie groups or from the Mostow list [23]. R E M A R K . If one omits the assumption '3-dimensional', the Mostow list shows that there is only one possibility where both properties are wrong, namely type III 5, including the affine action

02 f( d)d 0c Ot We apply this lemma to the horizontal action of K/R 2 o n ~ 2 = ¢l~vk{W}" Since Er fixes a point w e W and EEyI acts freely on ~ \ { Y, W}, the stabilizer (K/NZ)r of the horizontal action fixes no further line Y'~ v, Y' ~ Y, W. By the lemma the c o m m u t a t o r group of K/[R 2 a c t s transitively on R 2 = ~ \ { W} and is commutative (being the 2-dimensional c o m m u t a t o r of a 3-dimensional Lie group). So we have constructed a normal subgroup A of Z acting regularly on P\{ W}. This group A is a central extension of the translation group N2 by the group ~2 and is therefore nilpotent and homeomorphic to N4. There are only two possibilities for A: A = Nil x R or A -_ R 4. In the first case the centre C of Nil is contained in the translation group ~2 since the extension is central. The group C coincides with the c o m m u t a t o r subgroup A' of A and is therefore normal in E. But as we have seen above, the group ~ induces a transitive action o n P l ( ~ 2 ) , hence there cannot exist a 1-dimensional invariant subspace C in E2, a contradiction. The other case A ~ E4 also leads to a contradiction by (2). (9) Assume dim Z~rI = 0. Here the group Y o, 0 e PkW, acts as a 3-dimensional automorphism group on the translation group EEv,vr Since L2 has only a 2-dimensional automorphism group, this implies ZEv,vI = R 2. Again we consider the h o m o m o r p h i s m q~: Z / R 2 ~ GL2(R), which has now a 3-dimensional image consisting of all triangular matrices {(ca d)la, d > 0, c ER}. The kernel of (p is K/R 2 with d i m K = 4, and K/E 2 acts regularly on ~,\{W}. The horizontal action is therefore an RNS-action with regular normal subgroup K / ~ 2. Suppose K / ~ 2 ~ L2, then the horizontal action could be of dimension 4 only since L 2 has only a 2-dimensional automorphism group. This contradiction shows

K/[~2

_----- ~ 2

and

K ~ ~4, Nil x R or

o I xp (iolO)ot.

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Since the last group has only a 1-dimensional centre and since the group ~4 is excluded by (2), we need only study the case K ~ Nil x ~. Since the sum of two nilpotent ideals in a Lie algebra is nilpotent [,12, I. Prop. 6] there exists a unique maximal nilpotent ideal, called the nilpotent radical (or only nilradieaI). We denote by 91 the nilradical of the Lie algebra ~(E) of E. (a) dim91 = 5. Proof The nilradical 91 contains the 4-dimensional nilpotent ideal ~(K) of :~(Z). Since the commutator algebra ~(Z)' is nilpotent by [12, II.7], the nilradical 91 also contains ~(E)'. But from the action of Z on ~v\{ W} it follows that ~(Z)' s not contained in ~(K), hence dim 91/> 5. If dim 91 >~ 6, then 9l/~ z is a nilpotent ideal of ~(E)/~ 2 of dimension at least 4. But the horizontal algebra ~(E)/~2 has a nilradical of dimension 3, a contradiction. (b) The Lie algebra 91 has the following structure: 91 = , [~, ~] = ~, [~, i ] = ~, [~, ~3 = u, [~, u ] = ~.

Proof By construction ~(K) can be described by basis vectors t, t), u, u with [,~, t~] = D as the only non-trivial commutator. Here, corresponds to Ztv1. Since on the translation group there is induced the affine action III 7, we can choose a fifth basis vector ~ such that [~, u] = v and [,~, v] = 0. Note that ~ corresponds to the commutator subgroup of Zo. From the horizontal action we get the relations [~, ~] = t) + au + bu, [,~,t)] = cu + du, a, b, c, d e N. Assume first c = 0. Then the commutator E; has a 2-dimensional centralizer since [~, ~] = 0 and [~, t) - du] = 0. This implies that the oneparameter group E; fixes a quadrangle in contradiction to the lemma on quadrangles. From this contradiction follows c ~ 0, and by changing the basis suitably we can get c = 1, d = 0, a = b = 0. REMARK. There are six 5-dimensional nilpotent real Lie algebras which are not products of smaller algebras [-221, [8]. The Lie algebra 91 is characterized by the dimensions (5, 3, 2, 1) of the descending central series and by having a 3-dimensional maximal abelian ideal. (c) The group Z induces by conjugation the linear action

i,xpr

on ~3 = {(y, u, v)[y, u, ve ~}.

Proof Since exp 9l is normal in E, the group induced by exp 91 must be

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normal in the group induced by Z. From the Lie algebra of 91 we see that exp 9l induces on Ra the linear group

1

s

) o} S, Z E

.

1

Calculating the normalizer in the group of triangular matrices gives the group in (c). Since Z leaves the v-axis and the (u, v)-subspace invariant and since Z acts faithfully on ~3, the group induced by Z coincides with this normalizer. Note that Z is not the semidirect product of N3 with this 4-dimensional linear group. We have only a semidirect product of R 3 by

exp r

~\

s2/2

s

r, s, t e exp

describing the subgroup of N which fixes x -- 0. We now study the action of £ on W\{v} and show (d) The effective action of Z/Z[wl on W\{v} is transitive and isomorphic to the affine action G = N2>~ {(~ d)la, d > 0, c e JR}. Proof Assume by contradiction that 2 does not act transitively on W\{v}. Since Z cannot fix a point by (1) there would exist a 1-dimensional Z-orbit on W\{v}. Fixing a point w of this orbit would give a 6-dimensional group Z wThis group acts transitively on £~\{W} as an affine group. Since Ew is solvable we get dim 2[w] >~ 1. Using the assumption dim Z[rl = 0, this implies that the translation group with respect to the axis W is 3-dimensional and isomorphic to N3. By the structure of Z this must be the group ~3 = {(y, u, v) [ y, u, v ~ R}. The linear group

f( xpr exp/lo31 of ~3 is then a group of central collineations with centre o and axis W, and this would give a translation plane by the theorem on homologies and elations. The same argument shows that dim Z~w~ = 2, and therefore we have Ztvj = Erv,vJ = EEw,wI = Ztw]. So the group E/Etw ] = E/Et~ 1acts as an effective and transitive transformation group on ~v\{W} and simultaneously on

w\{v). If we introduce (4, r/)-coordinates on W\{v} such that the coordinate lines

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~ = const are lines of imprimitivity for the action of Z on W\ {v} then to every map

(x, y)~--~(ax + m, dy + cx + n), (a, c, d, m, n)eG on Ev\{W} there corresponds a map

(~, ~l)v--~(a'~ + m', d'tl + c'~ + n'), (a', c', d', m', n')eG on W\{v}, where ~ = ( )' is some automorphism of G. We consider W\{v} and also £,\{W} as 2-dimensional affine planes and use there the terms points and lines. Of course, these notions must not be confounded with points and lines of the original plane. (3) The automorphism ~ of G maps stabilizers of points in the action (£~\{W}, E/Eta1) to stabilizers of lines in the action (W\{v}, Z/Ztwj). The subgroup of Z which f x e s some x =-const. in ~ \ { W } acts transitively on W \ { v}. The subgroup of E which fixes an element (x, y) E ~2~\{W} and some line

y = rx + s through (x, y) alsofixes some point w e W\{v}. Proof A calculation in the affine group

shows that the stabilizers of points together with the stabilizers of nonvertical lines form a full orbit under the automorphism group of G. Assume that the stabilizer ZIx,y) also fixes some point (~, t/) s W\{v}. Then fixing the point (x, y, 0, 0), a further line S~x,y) and a further point (~, t/') would give a contradiction to the lemma on quadrangles. It follows that the stabilizer Z(x,r) fixes some line t / = r~ + s on W\{v} and acts transitively on this line. Together with the action of the group (x, y) ~ (x, y + n), (~, t/) ~ (~, q + n'), n s N, this proves the second statement of(e). Take now the stabilizer of Z/Z H on a line S~x.y), say (x, y) = (0, 0). This group is isomorphic to N x L 2 and induces on the set of lines {y = kx, k e R} through (0, 0) an L2-action which has the centre of N x L 2 as kernel. The related action on the corresponding fixed line in W\{v} is also an Lz-action with the same kernel. Therefore the stabilizer of a line y = kx through (0, 0) also fixes some point w e W\{v}. Recall that the group K acts regularly on P\W. The orbits under the group ~3 = {(y, u, v) ly" u. v ~ ~} define a fibration of the point space P \ W and the set of fibres is coordinatized by x ~ R. By applying the group R 3 to the origin 0 = (0, 0, 0, 0) we identify the fibre x = 0 with the group R 3. (f) For every proper line L~ v the intersection with the fibre x = 0 is a 1-

dimensional affine subspace of R 3. Proof The subgroup of Z fixing the fibre x = 0 acts transitively on E \ ~ ,

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by (e). Since this subgroup acts as an affine group on the fibre x = 0 by (c), it suffices to prove (f) for one line Lkv. Take the 3-dimensional stabilizer Eo and fix the x-axis in ~v\{W}. This gives the subgroup

exp r

r, t e exp

of Eo. By (e) this group fixes a point w e W\{v} and therefore fixes the line L = 0 v w. The intersection of L with the fibre x = 0 is some curve which must be invariant under the 2-dimensional group above. This implies that the intersection coincides with the y-axis. (g) The plane is a translation plane. Proof The group Z o contains the group

expr

re exp

of homotheties of ~3. Since the 1-dimensional subspaces of ~3 not in S~o,o) correspond to the lines of ~o\{S(o,o)}, the group

Itexp expr is a group of central collineations of the plane with centre o and axis W. Using the theorem on homologies and elations we now get a translation plane.

2.2. Case dimEtv I = 3 (10) The dual plane has a 3-dimensional translation group with axis W. Proof. By dualizing we get the assumptions dim Etw I = 3 and Z transitive on W\{v}. We apply the theorem on homologies and elations to the group fl = (Etwl) 1 with axis W." if Z \ W consists of one center 0~ IV,, then the line Y = 0 v v is fixed by E in contradiction to (1). If Z \ W consists of at least two points, then the elation group ~tw,wl is connected and has positive dimension. Furthermore, Z \ W is an orbit under f~tw,wl and is fixed by E. If dim ~[w,wl = 1 or 2, all elations in f~tw,w~have centre v since Z is transitive on Wk{v} (use the theorem on elations). Then the orbit is contained in a line Y~v, Y ~ W which is fixed under Z, a contradiction to (1). It follows that dim f~[w,wj = 3 and we have a plane with a 3-dimensional translation group.

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Such planes were studied in [6]. The reader is advised to study the methods of [-6], which will be used in the sequel. Denote by 3~ the set of all orbits under the translation group ~3 on P\W. Then X is homeomorphic to ~ [6] and we assert (11) Y~acts transitively on Y.. Proof Otherwise one orbit would be fixed under Z. We identify this orbit with the translation group Ztw I ~ R 3 and coordinatize it by {(y, u, v) l y, u, v e ~}, where Nz = {(u, v) [ u, v s N} denotes the translation group in direction v. The space • of orbits is coordinatized by x e N, where we take x = 0 for the fixed orbit. Note that the 1-dimensional translation groups with directions w e W, w ¢ v, correspond to the 1-dimensional subspaces of ~3 = {(y, u, v)ly, u, v ~ ~} complementary to ~2 = {(u, v) l u, v ~ ~}. The group Y. is the semidirect product of the translation group R 3 and the 4-dimensional stabilizer Zo, 0eN3, compare also [6, L e m m a 3]. We denote by A the subgroup of Z o which fixes every line S~o.y), y e N. This means for the corresponding 3 x 3 matrices acting on the translation group R3 = {(y, u, v)[y, u, v} that they have a '1' at position (1, 1). Therefore these matrices describe also the affine action of A on W\{v}. We have either dim A = 4 or dimA = 3. In the first case, the group A acts 2-transitively on W\{v}, otherwise we would get a contradiction to the lemma on quadrangles. Since dim Ztwj = 3, the action of A on W\{v} is 4-dimensional and therefore isomorphic to the group of similitudes. Taking the c o m m u t a t o r gives the linear group

1

m, neN

0 acting on R 3. Together with the translation group in direction v we get a subgroup

/(! 1 m

1

1) u,v,m,n~

n 0 of E isomorphic to ~4 in contradiction to (2). If dim A --- 3 then A acts transitively on W\(v} and we apply the lemma on free stabilizers. Either the stabilizer A w, we W\{v}, fixes some further point w' E W\{v}, then we get a contradiction to the lemma on quadrangles. Or the

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c o m m u t a t o r subgroup A' acts transitively on W\{v}. Then, up to conjugation, A' has the form

A'=

f(' i)1 o} {(: )l °t m ~

rn, n~

or

A'=

1

1 m, ne

.

In the first case we again get a subgroup ~4 c Z o. In the second case we calculate the normalizer of A' in the g r o u p of triangular matrices and get

Zo =~[

m

expr m

r, s, m, n~

.

exp

C h o o s i n g m = n = 0 and r = s, we see that there is a 1-dimensional g r o u p

I( exprexpt/rO} of homotheties of the translation g r o u p ~3. Therefore the plane is isomorphic to a plane studied in [6], [35]. But these planes are either dual translation planes or they have dim Y~ ~< 6. N o w the p r o o f of (11) is finished. F r o m (11) it follows that 2; acts transitively on ~v\{W}. Since we have already settled the case dim Z[v] = 2, we can assume dim 2[~] = 3, that is, we have a 3-dimensional g r o u p of shears with respect to the centre v. Denote by ~ the space of orbits of this g r o u p on W\{v}. Then we have a selfdual situation: there is a 3-dimensional translation g r o u p and I; is transitive on 3£, and dually there is a 3-dimensional g r o u p of shears and 12 is transitive on ~. We show that the transitive action of I2 on W\{v} is an affine action. To prove this we give a representation of this action as an affine action on PI(~a)\PI((U,V)): we m a p every point we W\{v} to its corresponding 1dimensional s u b g r o u p w * = 12[w,w~of the translation g r o u p R3; and we m a p every a ~ X to the linear a u t o m o r p h i s m ~ = (z ~ a-lza) of R 3. In this way we get an isomorphism of transformation groups (q~, a): (W\{v}, E ) ~ (PI(~3)\(PI({u, v)), X'). Dually the g r o u p Z acts on ~ \ { v } in an affine way. We choose horizontal coordinates {(x, y)[x, y ~ ~} for the affine action on ~ \ { W } such that the coordinate lines x = const, correspond to the elements of 3E. Dually we choose coordinates {(~, t/)[ ~, q ~ ~} on W\{v} such that the coordinate lines = const, correspond to the elements of ~. Both affine actions are of dimension 4.

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We can now improve statement (11) and show (12) E acts transitively on • x ~. Proof. Otherwise there would be a 1-dimensional orbit and we could fix two elements (xl, ~1) and (x2, ~2), x2 ~ xl, ~2 4: ~l, of this orbit by a 5dimensional subgroup of Z. Fixing, moreover, some line S~x,.yo~ ~v\{W} and some point (~x,th)~W\{v} would give a 3-dimensional group. Since the actions on ~v\{W} and on W\{v} are affine, this group would fix a further line Scx.y), (x, y) ~ (Xx, Yl), and a further point (~, t/) ~ (41, th) by the following LEMMA. If a one-parameter subgroup of GLz(~) fixes some line L through the origin and a parallel line 12, then besides the origin another point is fixed. Proof The subgroup of GL2(~ ) which fixes the line L' has connected component L z. If the given one-parameter subgroup is the normal subgroup of L/, then it is the group of shears along L and all points of L are fixed. If it is a subgroup complementary to the normal subgroup, then it fixes a line K~ 0, K ~ L and also the point K c~ L'. From this 3-dimensional group we can construct a contradiction to the lemma of quadrangles and (12) is proved. As an immediate consequence of (12) we get:

The 3-dimensional stabilizer Eo, O~ P\W, acts transitively and effectively on W\{v} (and the dual statement). Proof The subgroup of E fixing the element x = 0 of • acts transitively on by (12). Since the translation group ~3 acts trivially on W,, the group Zo acts transitively on $ also. Together with the group of shears in Z 0 we get the transitivity of E 0 on W\{v}. The effectivity of this action follows from the theorem on homologies and elations (example). (13) The induced action of Z on X × ~ has dimension 3. The kernel of ineffectivity is the normal suboroup 1-I generated by all translations with respect to the axis W and all shears with respect to the centre v. Proof Look at the action of Z on • and the action on ~. Both actions are transitive and may be isomorphic to the action R or the action Lz. Assume by contradiction that both actions were R-action. Then the 1-dimensional stabilizer fixing a point O ~ P \ W and a point w e W\{v} provides a contradiction to the lemma of quadrangles (use the lemma in (12)). Therefore one of the two actions is an L2-action and the induced action on 3Ex ~ has dimension at least 3. But since the 4-dimensional group 1-I acts trivially on 3~ x ~, the dimension is exactly 3. Note that FI is the projective group of the plane since there exist no homologies. The group 1-I is the semidirect product of the translation group

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~3 = {(y, U, /)) [ y, u, v e N} and the group of shears

1

n~N

0

1

with respect to the line S~o,o). This is a 4-dimensional nilpotent group. Let 91 be the nilradical of the Lie algebra ~2(Z), then (14) 5 ~< dim 91 ~< 6. Proof The c o m m u t a t o r algebra ~/of !~(Y.) is nilpotent by [12, II.7]. By (13) !;/is not contained in !~(FI). Since the sum of nilpotent ideals is nilpotent we get dim91 >~ 4 + 1 = 5. Since by (13) the Lie algebra of Z has a quotient isomorphic to L 2, the Lie algebra !~(Z) is not nilpotent itself and we have dim 91 -G<6.

3. CASES d i m 9l = 5 AND d i m 91 = 6 (15) Case dim91 = 5. Then up to duality we have an R-action on 3£ and an Lz-action on 6 . The stabilizer on one element of 3¢ fixes all elements of 3£ and is a 6-dimensional normal subgroup of E. It contains the nilradical, which can be written as the semidirect p r o d u c t of the translation g r o u p and a 2-dimensional subgroup of

1

m , n, C ff

c

.

1

Up to isomorphism there are three possible groups of this form given by c = 0 or m = 0 or c = m. If c = 0 we get the subgroup

m

1

n

0

u,v,m, nEN 1

of Y,, which is isomorphic to N4. If m = 0, then the subgroup defined by n = 0 would fix the subspace (y, v) of N 3 elementwise, a contradiction to the lemma on quadrangles. Therefore we have to consider only the third case, that is the nilpotent g r o u p

N = ~a>~

1 c

n, c~ I

.

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BETTEN

We let 22 act by conjugation on the translation group ~3. The image of this homomorphism (p: Y ~ GL3(R ) is a 4-dimensional linear group, which has

1

c, n E N

c

as a normal subgroup (since N is normal in Y~).Calculating the normalizer of this group in the group of lower triangle regular 3 x 3 matrices we get the group s)]

{exp(2r-s)

l:

,.°xP" exp

~}

.

If we choose r = s, m = n = 0, we get a one-parameter group of homotheties of ~3. Since the points w ~ W \ { v } correspond to 1-dimensional subspaces of N3, this would imply that 22 acts on W with a 4-dimensional kernel, a contradiction. REMARK. The group Z is an extension of the translation group N3 by a 4dimensional group. But this extension might not split. For our reasoning we need not know the structure of the group 22; we only use the linear action of E/R a on N3 induced by conjugation. (16) Case dim 9l = 6. In the list of all real 6-dimensional nilpotent Lie algebras [22], [34] there are only two types which do not contain a N4-subalgebra namely the algebras with the following commutator relations (which are written simply as product): Morozov No. 21: e l e 2 Morozov No. 22: e l e 2

= e 3 , e l e 5 = e 6 , e 2 e 3 ---- e 4 , e 2 e 4 =

es,

e 3 e 4 ---- e 6

and

---- e3, e l e 3 ---- e s , e l e 5 ---- e6, e 2 e 3 ---- e4, e 2 e 4 ---- es,

e 3 e 4 = e 6.

Assume that 9l is isomorphic to one of these algebras, then from the relations we get 9l' -- {@3, e4, es, e6), e3e4 = e6}. Since exp 9l induces an R-action on 3~ and on ~, it follows that 9l' c ~(H). But both of these algebras are 4dimensional, therefore ~ ' = ~(II). The centre is @5, e6) and coincides with the Lie algebra of the translation group Ztw,w k We now look for an ideal of~R which is isomorphic to R 3, is contained in the commutator algebra ~R' and contains @5, e6). Such an ideal must have the form (ce 3 + de,,, e 5, e6), e, d ~ R, (c, d) ~ (0, 0). It must contain the element [e2, ee3 + de4] = ce4 + des, which implies ee4 = k(ce3 + de4) for some k ~ R. From this follows c = 0 and

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we get the unique ideal (e4, es, e6). But from the geometric situation we k n o w that there must be two ideals of this kind, one corresponding to the translation group and the other corresponding to the group of shears with centre v. So in all cases we have either derived a subgroup ~4 or we have got a contradiction. Therefore the theorem is proved.

REFERENCES 1. Betten, D., '4-dimensionale Translationsebenen', Math. Z. 128 (1972), 129-151. 2. Betten, D., '4-dimensionale Translationsebenen mit kommutativer Standgruppe', Math. Z. 154 (1972), 125-141. 3. Betten, D., 'Transitive Wirkungen auf F1/ichen', Vorlesungsskript, Kiel, 1977. 4. Betten, D., 'Komplexe Schiefparabelebenen', Abh. Math. Sere. Univ. Hamburg 48 (1979), 7688. 5. Betten, D., 'Zur Klassifikation 4-dimensionaler projektiver Ebenen', Arch. Math. 35 (1980), 187 192. 6. Betten, D., '4-dimensionale projektive Ebenen mit 3-dimensionaler Kollineationsgruppe', Geom. Dedicata 16 (1984), 179-193. 7. Betten, D. und Knarr, N., 'Rotationsflfichen-Ebenen', Abh. Math. Sem. Univ. Hamburg 57 (1987), 227-234. 8. Dixmier, J., 'Sur les representations unitaires des groupes de Lie nilpotentes. III', Canad. J. Math. 10 (1958), 321-348. 9. Forst, M., 'Effektive Lie-Algebren-Paare der Codimension 2', Diplomarbeit, Kiel, 1977. 10. H/ihl, H., 'Homologies and elations in compact, connected projective planes', Top. Appl. 12 (1981), 49-63. 11. Hughes, D. R. and Piper, F. C., Projective Planes, Springer, New York, Heidelberg, Berlin, 1973, 1982. 12. Jacobson, N., Lie Algebras, Interscience, New York, 1962. 13. Knarr, N., 'Topologische Differenzenflfichenebenen', Diplomarbeit, Kiel, 1983. 14. Knarr, N., '4-dimensionale projektive Ebenen vom Lenz-Barlotti-Typ II.2', Results Math. 12 (1987), 134-147. 15. Knarr, N., 'Topologische Differenzenfl/ichenebenen mit nichtkommutativer Standgruppe', Dissertation, Kiel, 1986. 16. Knarr, N, '4-dimensionale projektive Ebenen mit grol3er abelscher Kollineationsgruppe', or. Geometry 31 (1988), 114-124. 17. Lie, S., Vorlesungen iiber kontinuierliche Gruppen, Teubner, Leipzig, 1893; Nachdruck: Chelsea Publ. Co., Bronx, New York. 18. Lie, S. und Engel, F., Theorie der Transformationsgruppen I - I I I , Teubner, Leipzig, 1888-93; Nachdruck: Chelsea Publ. Co., Bronx, New York. 19. L6wen, R., 'Four-dimensional compact projective planes with a nonsolvable automorphism group', Geom. Dedicata 36 (1990), 225-234. 20. L/ineburg, M., 'Kompakte, 4-dimensionale projektive Ebenen mit 8-dimensionaler Kollineationsgruppe', Diplomarbeit, T/ibingen, 1988. 21. Montgomery, D. and Zippin, L., Topological Transformation Groups, Interscience, New York, 1955. 22. Morozov, V. V., 'Klassifikation nilpotenter Lie Algebren der 6. Ordnung', Izv. Vyss. Uchebn. Zaved. Matematika 4(5) (1958), 161-171.

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23. Mostow, G. D., 'The extensibility of local Lie groups of transformations and groups on surfaces', Ann. Math. 52 1950, 606-636. 24. Pickert, G., Projektive Ebenen, 2. Auftage, Springer, Berlin, Heidelberg, New York, 1975. 25. Plaumann, P. und Strambach, K., 'Zweidimensionale Quasik6rper mit Zentrum', Arch. Math 21 (1970), 455-465. 26. Sagle, A. A. and Walde, R. E., Introduction to Lie Groups and Lie Algebras, Academic Press, New York and London, 1973. 27. Salzmann, H. R., 'Topological planes', Adv. in Math. 2 (1967), 1-60. 28. Salzmann, H. R., 'Kollineationsgruppenkompakter 4-dimensionaler Ebenen', Math. Z. 117 (1970), 112-124. 29. Salzmann, H. R., 'Kollineationsgruppen kompakter 4-dimensionaler Ebenen II', Math. Z. 121 (1971), 104-110. 30. Salzmann, H. R., 'Kompakte 4-dimensionale projektive Ebenen mit 8-dimensionaler Kollineationsgrupe', Math. Z. 130 (1973), 235-247. 31. Salzmann, H. R., 'Elations in four-dimensional planes', Gen. Top. and Appl. 3 (1979), 121-124. 32. Tits, J., 'Sur certaines classes d'~spaces homog~nes de groupes de Lie', Acad. Roy. Belg. Mere. Cl. Sci., Tome 29, Fasc. 3, 1955. 33. Tits, J., Liesche Gruppen und Algebren, Vorlesung, Bonn, 1965; Springer, Berlin, Heidelberg, New York, Tokyo, 1983. 34. Vergne, M., 'Vari6t6s des alg6bres de Lie nilpotentes', Th6se, Paris, 1966. 35. Weigand, C., 'Konstruktion topologischer projektiver Ebenen, die keine Translationsebenen sind', Mitteil. Math. Sere. Giel3en 177 (1987). A u t h o r ' s address:

D i e t e r Betten, M a t h e m a t i s c h e s S e m i n a r der Universitfit, L u d e w i g - M e y n - S t r . 4, 2300 Kiel,

F.R.G. (Received, January 8, 1990)