On two nonbuilding but simply connected compact Tits geometries of type $C_3$

Abstract

A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been obtained by Kramer and Lytchak (2014; 2019). According to their main result, all such geometries but two are quotients of buildings. The two exceptions are flat geometries of type $C_3$ and arise from polar actions on the Cayley plane over the division algebra of real octonions. The classification obtained by Kramer and Lytchak does not contain the claim that those two exceptional geometries are simply connected, but this holds true, as proved by Schillewaert and Struyve (2017). Their proof is of topological nature and relies on the main result of (Kramer and Lytchak 2014; 2019). In this paper we provide a combinatorial proof of that claim, independent of (Kramer and Lytchak 2014; 2019).