16 - dimensional compact projective planes with 3 fixed points

Let P 1⁄4 ðP;LÞ be a topological projective plane with a compact point set P of finite (covering) dimension d 1⁄4 dimP > 0. A systematic treatment of such planes can be found in the book Compact Projective Planes [15]. Each line L A L is homotopy equivalent to a sphere Sl with l j 8, and d 1⁄4 2l, see [15] (54.11). In all known examples, L is in fact homeomorphic to Sl. Taken with the compact-open topology, the automorphism group S 1⁄4 AutP (of all continuous collineations) is a locally compact transformation group of P with a countable basis, the dimension dimS is finite [15] (44.3 and 83.2). The classical examples are the planes PK over the three locally compact, connected fields K with l 1⁄4 dimK and the 16-dimensional Moufang plane O 1⁄4 PO over the octonion algebra O. If P is a classical plane, then AutP is an almost simple Lie group of dimension Cl, where C1 1⁄4 8, C2 1⁄4 16, C4 1⁄4 35, and C8 1⁄4 78. In all other cases, dimSc 12Cl þ 1c 5l. Planes with a group of dimension su‰ciently close to 12Cl can be described explicitly. More precisely, the classification program seeks to determine all pairs ðP;DÞ, where D is a connected closed subgroup of AutP and bl c dimDc 5l for a suitable bound bl d 4l 1. This has been accomplished for lc 2 and also for b4 1⁄4 17. Here, the case l 1⁄4 8 will be studied; the value of bl varies with the configuration of the fixed elements of D. Most theorems that have been obtained so far require additional assumptions on the structure of D. If dimDd 27, then D is always a Lie group [12]. By the structure theory of Lie groups, there are 3 possibilities: (i) D is semi-simple, or (ii) D contains a central torus subgroup, or (iii) D has a minimal normal vector subgroup, cf. [15] (94.26). The first two cases are understood fairly well: