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
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).
