16-dimensional compact projective planes with a large group fixing two points and two lines

Abstract

H e r r n P r o f e s s o r O t t o H . K e g e l z u m 7 0. G e b u r t s t a g Abstract. We determine all planes having the properties of the title with a group of dimension at least 33.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 [21]. Each line L ∈ L is homotopy equivalent to a sphere S with | 8 , and d = 2 , see [21, (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 [21, (44.3)]. The covering dimension dim Σ is an important parameter for characterizations of such planes. (For readers which are more familiar with the inductive dimension, we remark that for a locally compact group Λ, the inductive dimension ind Λ coincides with dim Λ and with the dimension of the connected component Λ 1 , cf. [21, (93.5, 6)].The classical examples are the planes P K over the 3 locally compact, connected fields K with = dim K and the 16 -dimensional Moufang plane O = P O over the octonion algebra O . If P is a classical plane, then Aut P is an almost simple Lie group of dimension C , where C 1 = 8, C 2 = 16, C 4 = 35 , and C 8 = 78 .In all other cases, dim Σ 1 2 C + 1 5 . Planes with small groups abound, those with a group of dimension sufficiently close to 1 2 C can be described explicitly. More precisely, the classification program seeks to determine all pairs (P, ∆) , where ∆ is a connected closed subgroup of Aut P and b dim ∆ 5 for a suitable bound b 4 − 1 . This has been accomplished for 2 and also for b 4 = 17 . Results in the case = 8 are as yet less satisfactory, and it is this case that will be considered here. (2000): 51H10, 51A35, 12K99. Mathematics Subject Classification

Compact planes, mostly 8-dimensional. A retrospect

“…Remarks. (1) An analogous theorem for 16-dimensional planes has been proved in [30]. In particular, distorted octonions have been defined in the same way.…”

“…All 16-dimensional planes with a group of dimension ≥ 35 have been described, provided the group does not fix exactly one point and one line ( [30], [31]). By [80] 83.26, a group of dimension > 30 fixes at most some collinear points and their join or two points and two lines.…”

16-dimensional compact projective planes with 3 fixed points

“…It is well-known that α Δ α is contained in the group Τ of translations with axis W and that α inverts each translation in T. Consequently, Υ acts faithfully on each invariant subgroup of T. There is only one irreducible representation of Υ in dimension *ξ!6, viz. the natural one on IR 8 . 1 7) shows that 0>^ (9. 9) Consider now the second case mentioned at the end of 7).…”

“…If dim Δ ^ 40, then 9 and its dual are translation planes [15] (87.7). All translation planes with dimA ^ 38 are described in [15] A method to construct all planes with exactly 2 fixed points have been given in [8].…”

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

“…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.…”