Exercise 19, § 1

Let A be an apartment. Assume that, for any two distinct neighbouring chambers C₁ and C₂, there is a folding (Exerc. 18) of A taking C₁ to C₂. Let C be a chamber of A and (C_i)_{i∈I} the family of chambers neighbouring C and distinct from C. Denote by s_i the reflection with respect to the panel C ∩ C_i (Exerc. 18 b)). Put S = {s_i | i ∈ I} and denote by W the group of automorphisms of A generated by the s_i.

a) Show that, for any chamber C', there exists w ∈ W such that C' = w(C) (argue by induction on d(C, C')).

b) Show that (W, S) is a Coxeter system (for i ∈ I, put

where $\pi_i$ is the folding taking C_i to C, and show that the assumptions of Prop. 6 are satisfied: to prove condition (C'), remark that if w ∈ P_{s_i} and ws_j ∉ P_{s_i}, then

Since w(C) and ws_j(C) are neighbouring, it follows that s_i = ws_j w^{-1} (Exerc. 18 b)).

c) Let F be a facet of the chamber C. Show that the stabilizer W_F of F in W is generated by the s_i ∈ S such that F ⊂ C ∩ C_i (let w ∈ W_F with l_S(w) > 1 and let i ∈ I be such that w = s_i w' with l(w') = l(w) - 1; by Prop. 6, w' ∈ P_{s_i}, hence w(C) ⊂ s_i \pi_i(A), F ⊂ \pi_i(A) ∩ s_i \pi_i(A) and s_i ∈ W_F). In particular, w(C) = C if and only if w = 1.

d) Show that the map a ↦ W_{\{a\}} is an isomorphism from the apartment A to the apartment associated to (W, S) (Exerc. 16), compatible with the action of W.