16-dimensional compact projective planes with a collineation group of dimension >= 35

Arch. Math. 90 (2008), 284–288 c 2008 Birkh¨
auser Verlag Basel/Switzerland 0003/889X/030284-5, published online 2008-02-20 DOI 10.1007/s00013-007-2466-y

Archiv der Mathematik

16-dimensional compact projective planes with a collineation group of dimension ≥ 35 Helmut Salzmann Abstract. A compact projective plane is isomorphic to the classical Moufang plane if it admits an automorphism group of dimension at least 38 which fixes a non-incident point-line pair. Mathematics Subject Classification (2000). 51H10. Keywords. Compact planes, Moufang plane.

Let P = (P, L) be a topological projective plane with a compact point set P of finite (covering) dimension d = dim P > 0 . A systematic treatment of such planes can be found in the book Compact Projective Planes [11]. Each line L ∈ L is homotopy equivalent to a sphere S
with
| 8 , and d = 2
, see [11] (54.11). In all known examples, L is in fact homeomorphic to S
. Taken with the compact-open topology, the automorphism group Σ = Aut P (of all continuous collineations) is a locally compact transformation group of P with a countable basis, the dimension dim Σ is finite [11] (44.3 and 83.2). In many cases, Σ is known to be even a Lie group, cf. [11] (87.1) and [6]. Here, we deal exclusively with 16-dimensional planes; then the full automorphism group Σ is a Lie group whenever dim Σ ≥ 29 , see [9]. The last theorem in the book [11] states that a plane with dim Σ ≥ 40 can be coordinatized by a “mutation” of the octonions with a multiplication c ◦ z = (1−τ )cz +τ zc, where 21 = τ ∈ R . In the meantime, considerable progress has been made towards a classification of compact planes admitting a connected group ∆ of automorphisms with dim ∆ ≥ 35 . All such planes have been determined in which ∆ does not fix exactly one point and one line, see [2], [3] and the references given there. Here we will deal with the case that ∆ fixes exactly two non-incident elements. The following is already known and will be used:

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Theorem S. If ∆ is semi-simple, dim ∆ > 28 , and if ∆ fixes some element (point ∼ Spin (R, r) with r ≤ 1 , or P is the classical Moufang plane, or line), then ∆ = 9 see [4], [5]. It is not known whether or not there exist non-classical planes with a group Spin9 (R, r) . Theorem T. If ∆ contains a central torus, dim ∆ > 30 , and if ∆ fixes some element, then P is classical . This is a consequence of the result in [8]. Without any assumption on the structure of ∆ we will now show: Theorem. If the connected group ∆ ≤ Aut P fixes exactly one line W and one point a ∈ / W , and if dim ∆ ≥ 35 , then ∆ contains a group Spin9 (R, r) with r ≤ 1 and dim ∆ ≤ 37, or ∆ is triply transitive on W and P is the classical Moufang plane. The proof depends on B¨ odi’s result [1]: Stiffness Theorem. Assume that the connected group Λ fixes a quadrangle. Then Λ is isomorphic to the 14 -dimensional compact group G2 or dim Λ < 14 . If Λ is a Lie group and if the fixed elements of Λ form a connected subplane E , then Λ is isomorphic to G2 or its subgroup SU3 C or dim Λ < 8 . Moreover, dim E = 8 implies dim Λ ≤ 3, and Λ ∼ = G2 implies dim E = 2 . Lemma. Any non-trivial action of the group G2 on S8 is equivalent to the natural action of Aut O on O ∪ {∞}. In particular, G2 can act on R7 only in the standard way as a subgroup of SO7 R , see [11] (96.35). The action of the group G2 on S8 is described in detail in [11] (11.30–33); for SO9 R and Spin9 (R) see, e.g., [11] §18. Proof of the Theorem. The following cases will be distinguished: (A) ∆ is transitive on W , (B) ∆ is doubly transitive on some orbit V ⊂ W with 0 < k := dim V < 8 , (C) ∆ is said to have small orbits, if neither (A) nor (B) is true. If dim ∆ ≥ 40 , then ∆ contains a transitive translation group (and hence does not fix both a and W ), or P is classical (see [11] (87.7)). Because any quadrangle in the classical plane generates a connected subplane, the strong form of the Stiffness Theorem applies in this case. For dim ∆ < 40 , however, it is conceivable that there exist 0 -dimensional subplanes. Therefore, only the weak first part of the Stiffness Theorem can be used, or the connectedness of E has to be verified. Let u, v, w be suitable points of W and choose c on aw . Write Γ1 for the connected component of the (Lie) group Γ . Consider the stabilizers Ω = (∆u,v,w )1 and Λ = (Ωc )1 . The dimension formula [11] (96.10) yields (∗)

35 ≤ dim ∆ ≤ dim u∆ + dim v ∆u + dim w∆u,v + dim cΩ + dim Λ ≤ 46 .

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Arch. Math.

(A) If each orbit of ∆ on W is 8 -dimensional, then ∆ is transitive on W and ∆ has a compact subgroup Φ ∼ = Spin9 R of dimension 36 acting in the standard way as SO9 R on W ≈ S8 , see [11] (96.11a, 19, 21, 22, and 55.40). Conversely, Φ ≤ ∆ implies transitivity of Φ on W by [11] (96.13 (a) and (b)). From dim ∆ ≤ 40 it follows that ∆ is the product of Φ and its centralizer P. Assume that wP = w for some w ∈ W . Then Φc acts trivially on the subplane a, c, wP and dim Φc ≥ 20 . This contradicts the Stiffness Theorem. Hence P consists of homologies with axis W . The stabilizer Φv fixes a second point u ∈ W and dim Φu,v,c ≥ 12 . Since the fixed points of Φu,v,w on W form a connected subset of W , the Stiffness Theorem shows Φu,v,c ∼ = G2 . Therefore, the fixed points of this group form a 2 -dimensional plane E . Because P acts faithfully on E , we have dim P ≤ 1 and dim ∆ ≤ 37 . If dim ∆ > 40 , then P is classical and Σa,W is the product of Υ ∼ = Spin10 (R, 1) and a one-dimensional group of homologies, see [11] § 15 for the structure of Υ and note that Υ is triply transitive on W . It is well-known that Φ is a maximal compact subgroup of Υ , in fact, Φ is even maximal as an abstract subgroup, cf. [12 p. 160/1], [11] (94.34). Since dim(∆ ∩ Υ) ≥ 40 , it follows that Υ ≤ ∆ . (B) (1) Let v ∆ = V ⊂ W be a k -dimensional orbit with 0 < k < 8 . If the points u, v, w are chosen in V , then their orbits have dimension at most k and dim cΩ ≤ 8 . Hence (∗) implies 35 ≤ 3k + 8 + 14 and k ≥ 5 . (2) Suppose that ∆ is doubly transitive on V . All possibilities for the group ∆V induced by ∆ on V are known explicitly, see [11] (96.17). Either ∆V is simple and V is a sphere or a projective space, or ∆V is an extension of Rk ≈ V by a transitive linear group. (3) If V ≈ S7 , then a doubly transitive group on V is isomorphic to O9 (R, 1) , PSU5 (C, 1) , or PU3 (H, 1) . Since the kernel of the action of ∆ on V is at most 8 -dimensional, it follows that dim ∆V ≥ 27 , ∆V ∼ = O9 (R, 1) and ∆ is even triply transitive on V . By [11] (55.40), the group SO5 R cannot act on P . Hence ∆ contains a subgroup Ψ isomorphic to the 36 -dimensional covering group Spin9 (R, 1) of O9 (R, 1) . (4) Assume now that Ψ ≤ ∆ and that ∆ is not transitive on W . We will show that dim ∆ ≤ 37 . Let V be an orbit of ∆ of dimension dim V < 8 and choose u, v, w ∈ V . Then (∗) implies dim ∆ ≤ 3 · 7 + 8 + 14 = 43 . The representation of Ψ on the Lie algebra of ∆ shows that ∆ is an almost direct product of Ψ and its centralizer P = Cs ∆ Ψ . If wP = w for some w ∈ W , then Ψc induces the identity on the connected subplane a, c, wP and dim Ψc ≥ 20 , a contradiction to the Stiffness Theorem. Therefore P consists of homologies with axis W . Suppose that dim ∆ > 37 . Then dim Ψu,v,c ≥ 36 − 3 · dim V − 8 = 7 and Ψu,v,c acts trivially on the connected subplane F = u, v, cP . Consequently, dim F ≤ 4 . By [11] (55.4), the plane E = u, v, a, c is also connected and the strong form of the Stiffness Theorem applies. If dim ∆ ≥ 38 , then dim Λ ≥ 9 . Hence Λ ∼ = G2 and dim E = 2 . Now P acts faithfully on E as a group of homologies, dim P ≤ 1 and dim ∆ ≤ 37 .

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 (5) A doubly transitive group on V ≈ Sk , k < 7 , is isomorphic to POk+2 (R, 1) or to PSU4 (C, 1) . The hypothesis dim ∆ ≥ 35 leads to a contradiction in the same way as in the previous step.

(6) The cases where V is a real projective space can easily be excluded by (∗) . If k = 6 and V is homeomorphic to the complex projective space, then ∆V ∼ = PSL4 C and dim w∆u,v ≤ 2 for suitable w , moreover, a, c, u, v is connected. Now (∗) gives dim Λ ≥ 35 − 2k − 2 − 8 ≥ 13 , and the the Stiffness Theorem implies Λ ∼ = G2 , but this group is not contained in PSL4 C , a contradiction. (7) Let ∆ be doubly transitive on V ≈ Rk or C3 and choose u, w ∈ V . The argument of step (6) yields dim Λ ≥ 35 − 2k − 2 − 8 ≥ 11 and Λ ∼ = G2 . By the Lemma, Λ fixes only one point of V . Hence Λ cannot be contained in Ω . This is a contradiction. Note that the only possibility in case (B) is described in steps (3) and (4). (C) (1) We distinguish between classical and non-classical planes. First, let P be non-classical. Because of Theorems S and T combined with (A) and (B 4), we may assume that ∆ has a minimal normal subgroup Θ ∼ = Rt . From Theorem A ∼ in [10] it follows that Θ = R consists of homologies with axis W and center a. Let Γ denote the centralizer Cs Θ . Then dim Γ ≥ 34 and Γu,v,c acts trivially on the connected subplane u, v, cΘ . If P is classical, any quadrangle generates a connected subplane and we put Γ = ∆ . Since ∆ has small orbits, the points w and v may be chosen in such a way that 2 < dim v Γc < dim wΓ < 8 . If dim Γu,v,w > 16 , then the Stiffness Theorem implies (Γu,v,c )1 = Λ ∼ = G2 . This will lead to a contradiction as follows. Let H = (Γc )1 and U = v H . Then Λ acts on U and the Lemma implies that U contains an orbit z Λ ≈ S6 . Consequently, dim H ≥ 20 . The homogeneous space M = H/Hv is a 6 -dimensional connected manifold. Any orbit S ≈ S6 of Λ in M is open in M by domain invariance. Hence S = M ≈ U , but v Λ = v ∈ U . (2) By the last step, dim(Γ ∩ Ω) ≤ 16 and dim Ω ≤ 17 . On the other hand, dim Ω ≥ 16 , since ∆ is not doubly transitive on a 7 -dimensional orbit and dim ∆ ≥ 35 . ∼ SU3 C by step (1) (3) If dim Ω = 17 , then dim(Γ ∩ Ω) = 16 and K := Γ ∩ Λ = and the Stiffness Theorem. The fixed elements of K form a connected subplane F of dimension at most 4 , and a, c, u, v generate a connected subplane, see [11] (55.4). We have dim Λ = dim Ωc > 8 , hence Λ ∼ = G2 , but dim Λ − dim K ≤ 1 , a contradiction. (4) Now let dim Ω = 16 . This can only happen if dim w∆ = 7 and ∆w is doubly transitive on a 6 -dimensional orbit U ⊂ w∆ . The assumption dim Λ > 8 would lead to a contradiction as in step (1). Hence dim ∆c = 20 and ∆c acts effectively and still doubly transitively on U . However, there is no 20 -dimensional doubly transitive group on U : for compact U this follows from [11] (96.17); if U is not compact, then U ≈ R6 and a 2 -transitive group X satisfies dim X ≤ 16 or dim X ≥ 22 , see [11] (96.21,22).

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Thus, case (C) is impossible. This completes the proof of the theorem. References ¨ di, On the dimensions of automorphism groups of eight-dimensional ternary [1] R. Bo fields. II. Geom. Dedicata 53, 201 – 216 (1994). ¨hl and H. Salzmann, 16-dimensional compact projective planes with a large [2] H. Ha group fixing two points and two lines. Arch. Math. 85, 89 – 100 (2005). ¨hl and H. Salzmann, 16-dimensional compact projective planes with a large [3] H. Ha group fixing two points and only one line, to appear. [4] B. Priwitzer, Large semisimple groups on 16 -dimensional compact projective planes are almost simple. Arch. Math. 68, 430 – 440 (1997). [5] B. Priwitzer, Large almost simple groups acting on 16 -dimensional compact projective planes. Monatsh. Math. 127, 67 – 82 (1999). [6] B. Priwitzer and H. Salzmann, Large automorphism groups of 16-dimensional planes are Lie groups. J. Lie Theory 8, 83 – 93 (1998). [7] H. Salzmann, Compact 16 -dimensional projective planes with large collineation groups. Math. Ann. 261, 447 – 454 (1982). [8] H. Salzmann, Characterization of 16 -dimensional Hughes planes. Arch. Math. 71, 249 – 256 (1998). [9] H. Salzmann, Large automorphism groups of 16-dimensional planes are Lie groups. II, J. Lie Theory 9, 481 – 486 (1999). [10] H. Salzmann, On the classification of 16 -dimensional planes. Beitr. Alg. Geom. 41, 557 – 568 (2000). ¨hl, R. Lo ¨ wen, and M. Strop[11] H. Salzmann, D. Betten, T. Grundhofer, H. Ha pel, Compact Projective Planes. Berlin - New York: W. de Gruyter 1995 [12] J. Tits, Sur certaines classes d’espaces homog`enes de groupes de Lie. M´em. de l’Acad´emie Royale de Belgique, Classe de Sciences XXIX, Fasc. 3, 1955 Helmut Salzmann, Mathematisches Institut der Universit¨ at T¨ ubingen, Auf der Morgenstelle 10, D-72076 T¨ ubingen, Germany e-mail: [email protected] Received: 28 May 2007