Exercise 24, § 1

A structured building is a building $I$ equipped with a set $\mathfrak{U}$ of sub-buildings satisfying the following conditions:

(SB 1) The sub-buildings $A \in \mathfrak{U}$ are apartments.

(SB 2) Any two chambers of I are contained in at least one element of $\mathscr{U}$. (SB 3) Whenever $A_1, A_2 \in \mathscr{U}$ are such that $A_1 \cap A_2$ contains a chamber, there exists an isomorphism from $A_1$ to $A_2$ leaving fixed the points of $A_1 \cap A_2$.

Let $(I, \mathscr{U})$ be a structured building. The elements of $\mathscr{U}$ are called the apartments of $(I, \mathscr{U})$, or simply those of I.

a) Show that any two apartments of I are isomorphic. Let C be a chamber of I and A an apartment of I containing C. Show that there exists a unique endomorphism $\rho$ of I (called the retraction with centre C of I onto A) such that $\rho(a) = a$ for all $a \in A$ and that, for any apartment $A'$ containing C, the restriction of $\rho$ to $A'$ is an isomorphism from $A'$ to A (remark that, by (SB 2) and Exerc. 15 b), for any apartment $A'$ containing C, there exists a unique isomorphism $\rho_{A'}$ of $A'$ leaving all the points of C fixed). Show that $\rho^2 = \rho$ and that $\rho^{-1}(C) = C$.

b) Let A be an apartment of I, $C_0$ a chamber and F a facet contained in A. Let $(C_0, C_1, \ldots, C_n)$ be a gallery of smallest possible length such that $F \subset C_n$. Show that $C_i \subset A$ for $1 \leq i \leq n$ (argue by contradiction: if $C_i \subset A$ and $C_{i+1} \notin A$, consider the retraction of I onto A with centre the chamber of A distinct from $C_i$ and containing $C_i \cap C_{i+1}$).

c) Let A be an apartment of I, C a chamber of A, F a panel of C, $C'$ a chamber of I and $\Gamma = (C_0, \ldots, C_n = C')$ a gallery of smallest possible length such that $F \subset C_0$. Show that the retraction $\rho$ of I onto A with centre C transforms $\Gamma$ into a gallery

of smallest possible length such that $F \subset C'_0$ (consider an apartment $A'$ containing $C'$ and F and apply b) and the fact that the restriction of $\rho$ to $A'$ is an isomorphism).

d) Let A be an apartment, $C_1$ and $C_2$ two distinct neighbouring chambers contained in A, $C'$ a chamber containing $C_1 \cap C_2$ and distinct from $C_1$ and $C_2$, and $A'_i$ an apartment containing $C_i$ and $C'$ (for $i = 1, 2$). Let $\varphi_i$ (resp. $\psi_i$) be the retraction of I onto A (resp. $A'_i$) with centre $C_i$ (resp. $C'$) and let $\rho_i$ be the restriction of $\varphi_i \circ \psi_i$ to A. Let C be a chamber of A; show that if

then $\rho_1(a) = a$ for all $a \in C$ and $d(C, C_1) < d(C, C_2)$ (consider a minimal gallery $\Gamma$ with extremities C and $C_1$ and apply c) to show that $\rho_1(\Gamma)$ is minimal; then use Exerc. 15 b)). Show that, if $d(C, C_2) < d(C, C_1)$, then $\rho_1(C) \neq C$ and $\rho_1^2(C) = \rho_1(C)$ (take a minimal gallery $\Gamma$ with extremities C and $C_2$ and use c) to show that $\rho_1(\Gamma)$ is a gallery of smallest possible length having one extremity equal to $\rho_1(C)$ and the other containing $C_1 \cap C_2$; deduce that $d(C_1, \rho_1(C_1)) \leq d(C_2, \rho_1(C_1))$).

Show that $\rho_i$ is a folding (Exerc. 19) of $A$ (show that $A = \rho_1(A) \cup \rho_2(A)$) and define an involutive automorphism $\sigma$ of $A$ by putting $\sigma(a) = \rho_2(a)$ if $a \in \rho_1(A)$ and $\sigma(a) = \rho_1(a)$ if $a \in \rho_2(A)$; show that if $C$ is a chamber contained in $\rho_1(A)$, then $\rho_1^{-1}(C) = \{ C, \rho_2(C) \}$.

e) Let $I$ be a spacious structured building, that is to say that every panel is contained in at least three chambers. Show that there exists a Coxeter system $(W, S)$, unique up to isomorphism, such that the apartments of $I$ are isomorphic to the apartment $A_0$ associated to $(W, S)$ (use $d$ and Exerc. 19)). Let $A$ be an apartment of $I$ and $\varphi$ an isomorphism from $A_0$ to $A$. Show that there exists a unique numbering of $I$, with values in $S$, such that the types of $a$ and $\varphi(a)$ are equal for all $a \in A_0$ (choose a chamber $C$ of $A$ and show, by using $b$), that, if $A'$ and $A''$ are two apartments of $I$ containing $C$, the numberings of $A'$ and $A''$ extending that determined on $C$ coincide on $A' \cap A''$.

We shall say that the Coxeter system $(W, S)$ and the numbering thus obtained are adapted to $I$.

Show that the retractions introduced in a) are allowed endomorphisms. Show that a subset of $I$ contained in an apartment $A$ of $I$ is convex in $I$ if and only if it is convex in $A$ (Exerc. 20).

f) We retain the notation of e) and let $\mathscr{U}$ be the set of allowed isomorphisms from $A_0$ to the various apartments of $I$. Show that, if $\varphi, \psi \in \mathscr{U}$ and if $C$ is a chamber and $F$ a facet of $I$ contained in $\varphi(A_0) \cap \psi(A_0)$, there exists an element $w \in W$ such that $\psi^{-1}(C) = w \varphi^{-1}(C)$ and $\psi^{-1}(F) = w \varphi^{-1}(F)$ (consider an isomorphism $\lambda$ from $\varphi(A_0)$ to $\psi(A_0)$ leaving fixed the points of $\varphi(A_0) \cap \psi(A_0)$ and apply Exerc. 21 b) to the automorphism $\psi^{-1} \circ \lambda \circ \varphi^{-1}$ of $A_0$.