On exceptional homogeneous compact geometries of type $\mathsf C_3$

Abstract

We provide a uniform framework to study the exceptional homogeneous compact geometries of type C3. This framework is then used to show that these are simply connected, answering a question by Kramer and Lytchak, and to calculate the full automorphism groups.MSC 2010: 51E24, 57S15

On two nonbuilding but simply connected compact Tits geometries of type C3

“…Proof. Claims (1), (2) and (3) (This is essentially the same argoment as used by Schillewaert and Struyve to prove Lemma 6.6 of [11].) ✷…”

“…F and ⊥ instead of ⊥ F , for short. As usual, F * stands for the multiplicative group of F. Following Schillewaert and Struyve [11], we construct a C 3 -geometry Γ F (A) as follows.…”

“…Explicitly, As in the Introduction, let V 1 , V 2 and V 3 be the spaces defined on Γ 1 , Γ 2 and Γ 3 as above. It is straighforward to check that Γ {i,j} is closed in V i ×V j for every choice of 1 ≤ i < j ≤ 3 and the set of chambers Γ {1,2,3} is closed in As G acts flag-transitively on Γ, we can recover Γ as a coset-geometry from G. Comparing flag-stabilizers, it turns out that, when (F, A) = (C, O), the pair (Γ, G) is just the exceptional geometry considered by Kramer and Lytchak in [5] (see also Schillewaert and Struyve [11]). When (F, A) = (R, H) then (Γ, G) is the exceptional geometry of [6].…”

“…So, it is natural to ask if Γ is simply connected. The following theorem, due to Schillewaert and Struyve [11], answers this question in the affirmative:…”

“…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 [11]. The proof by Schillewaert and Struyve is of topological nature and relies on the main result of [5] and [6].…”