16-dimensional compact projective planes with 3 fixed points

Abstract

Let 9 = (P, £) 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 [15]. Each line L e £ is homotopy equivalent to a sphere §/ with / 1 8, and d = 2/, see [15] (54.1 1). In all known examples, L is in fact homeomorphic to §/. Taken with the compact-open topology, the automorphism group Σ = Aut^ (of all continuous collineations) is a locally compact transformation group of Ρ with a countable basis, the dimension dim Σ is finite [15] (44.3 and 83.2). The classical examples are the planes κ over the three locally compact, connected fields IK with ( — dim IK and the 16-dimensional Moufang plane G = έΡ® over the octonion algebra Θ. If 9 is a classical plane, then Aut^ is an almost simple Lie group of dimension Q, where C = 8, €2 = 16, €4 = 35, and Cg = 78. In all other cases, dim Σ < ^ Q + 1 < 5Λ Planes with a group of dimension sufficiently close to C( can be described explicitly. More precisely, the classification program seeks to determine all pair s (&, Δ), where Δ is a connected closed subgroup of Aut & and bf ^ dim Δ 5^ for a suitable bound be 4/ — 1 . This has been accomplished for t ^ 2 and also for 64 = 17. Here, the case ( = 8 will be studied; the value of bt varies with the configuration of the fixed elements ofA. Most theorems that have been obtained so far require additional assumptions on the structure of Δ. If dim Δ 27, then Δ is always a Lie group [12]. By the structure theory of Lie groups, there are 3 possibilities: (i) Δ is semi-simple, or (ii) Δ contains a central torus subgroup, or (iii) Δ has a minimal normal vector subgroup, cf. [15] (94.26). The first two cases are understood fairly well: