On two non-building but simply connected compact Tits geometries of type C3

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 (Homogeneous compact geometries, Transform. Groups 19 (2016), 43-58 and Erratum to: Homogeneous compact geometries, Transform. Groups, to appear). According to their main result, all such geometries but two are quotients of buildings. The two exceptions are flat geometries of type C3 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 (On exceptional homogeneous compact geometries of type C3, Groups Geome. Dyn. 11 (2017), 1377-1399). The proof by Schillewaert and Struyve is of topological nature and relies on the main result of Kramer and Lytchak. In this paper we provide a combinatorial proof of that claim, independent of Kramer and Lytchak's result.