Exercise 18, § 1

Let $A$ be an apartment (Exerc. 15). A folding of $A$ is an endomorphism $\pi$ of $A$ such that $\pi^2 = \pi$ and such that every chamber of $A$ is the image under $\pi$ of 0 or 2 chambers.

a) Let $\pi$ be a folding of $A$ and $C$ a chamber of $A$ such that $\pi(C) = C$. For any neighbouring chamber $C'$ of $C$, either $\pi(C') = C'$ or $\pi(C') = C$; if $a \in C$ we have $\pi(a) = a$. Show that, if $(C_0 = C, C_1, \ldots, C_n)$ is a gallery, either $\pi(C_i) = C_i$ for all $i$ or $(C_0, \pi(C_1), \ldots, \pi(C_n))$ is not minimal (and in fact has two consecutive equal chambers). Deduce that any minimal gallery whose extremities are invariant under $\pi$ is invariant under $\pi$. If $(C = C_0, C_1, \ldots, C_n)$ is a minimal gallery and if $\pi(C_n) \neq C_n$, there exists an index $i$ with $0 \leq i < n$ such that $\pi(C_j) = C_j$ for $0 \leq j \leq i$ and $\pi(C_j) \neq C_j$ for $i < j \leq n$.

b) Let $C_1$ and $C_2$ be two distinct neighbouring chambers and $\pi, \pi'$ two foldings of $A$. Assume that $\pi(C_0) = C_1$ and $\pi'(C_1) = C_0$. Let $C$ be a chamber, and consider the following three conditions:

Show that $(1) \implies (2) \implies (3)$ and deduce that the three conditions are equivalent. Show that $\pi$ (resp. $\pi'$) is the unique folding of $A$ taking $C_2$ (resp.

C₁) to C₁ (resp. C₂) (assume that (2) is satisfied and let (C₁, C₂', ..., Cₙ' = C) be a minimal gallery: show that π'(C'_{j+1}) is the unique chamber distinct from π'(C'_j) and containing the panel π'(C'_j ∩ C'_{j+1})). Show that π($\mathscr{C}$) and π'($\mathscr{T}$) form a partition of the set $\mathscr{T}$ of chambers of A and that π(a) = π'(a) = a for all a ∈ π(A) ∩ π'(A). Show that the map from A to itself that coincides with π' on π(A) and with π on π'(A) is an involutive automorphism of A. It is called the reflection with respect to the panel C₁ ∩ C₂. Show that it is the only non-trivial automorphism of A leaving fixed all the points of C₁ ∩ C₂ (use Exerc. 15 b)).

c) Suppose that A is the apartment associated to a Coxeter system (W, S) and retain the notation of Exerc. 16. Let C₁ and C₂ be two neighbouring chambers and let t be the element of T such that the wall L_t is the support of C₁ ∩ C₂. Let M_j be the half of A determined by L_t and containing C_j (for j = 1, 2). Show that the map π defined by π(a) = a if a ∈ M₁ and π(a) = t(a) if a ∈ M₂ is a folding of A such that π(C₂) = C₁ and that the reflection with respect to the panel C₁ ∩ C₂ is the map a ↦ t(a).