Exercise 13, § 2

We use the notation of Exerc. 24 of § 1: $(I, \mathscr{U})$ is a spacious structured building equipped with a Coxeter system $(W, S)$ and an adapted numbering and $\mathscr{J}$ is the set of allowed isomorphisms from the apartment $A_0$ associated to $(W, S)$ to the various apartments of $(I, \mathscr{U})$. Further, let $G$ be a group of allowed automorphisms of I, preserving E: the group $G$ then acts on $\mathscr{J}$ and we assume that $G$ acts transitively on $\mathscr{J}$.

Denote by C a chamber of I, A an apartment of $\mathscr{U}$ containing C, $\varphi$ an allowed isomorphism from $A_0$ to A taking the canonical chamber $C_e$ of $A_0$ to C, B the stabilizer of C in G and N the stabilizer of A in G.

a) Show that, if $A'$ and $A''$ are two apartments belonging to $\mathscr{U}$, and containing the same chamber, there exists $g \in G$ such that $g(A') = A''$ and that $g(a) = a$ for all $a \in A' \cap A''$. Show that $G$ acts transitively on the set of pairs $(A, C)$ where $A \in \mathscr{U}$ and C is a chamber of A.

b) Show that the map $n \mapsto \varphi^{-1} \circ n \circ \varphi$ is a surjective homomorphism from N to W, with kernel $B \cap N$. We can thus identify $N/(B \cap N)$ with W.

c) Show that conditions (WI 1) and (WI 2) of Exerc. 10 d) are satisfied (use the following fact: if an apartment of I contains two chambers $C'$ and $C''$, it contains every minimal gallery with extremities $C'$ and $C''$ (§ 1, Exerc. 24 b))).

d) Show that condition (G) of Exerc. 10 g) (and a fortiori condition (G$_0$) of Exerc. 11) is satisfied (with the notation of (G), consider an apartment $A'$ (resp. $A''$) of I containing $C_0$ and $C'$ (resp. $C''$); show by using Exerc. 24 b) of § 1 that $\Gamma \subset A'' \cap A''$ and use a)).

e) Show that $(G, B, N, S)$ is a Tits system and that $(I, \mathscr{U})$ is canonically isomorphic to the associated numbered structured building (Exerc. 12).