Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

Abstract

This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E6(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim \(\), see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim \(\), then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin9 (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL3?ċSpin3ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.