Exercise 15, § 1

⁷ Let $A$ be a set and $\mathscr{C}$ a subset of $\mathfrak{P}(A)$. The elements of $\mathscr{C}$ are called the chambers of $A$ and a subset $F$ of a chamber $C$ is called a facet, the codimension of $F$ in $C$ being the cardinal of $C - F$. A facet $F$ is said to be a panel of $C$ if $F$ is of codimension 1 in $C$. Two chambers $C$ and $C'$ are said to be adjoining

⁷ Exercises 15 to 24, as well as Exercises 3 to 17 of § 2, are hitherto unpublished and were communicated to us by J. Tits.

if they have a common panel F: then either C = C' or F = C ∩ C'. A gallery of length n is a sequence $\Gamma = (C_0, C_1, \ldots, C_n)$ of $n + 1$ chambers such that $C_i$ and $C_{i+1}$ are adjoining for $0 \leq i \leq n - 1$. Then $C_0$ and $C_n$ are called the ends of $\Gamma$. The gallery $\Gamma$ is called injective if $C_i \neq C_{i+1}$ for $0 \leq i \leq n - 1$ and is called minimal if there is no gallery with the same ends and length < n.

The set A (together with $\mathscr{C}$) is a building if every element of A belongs to at least one chamber and if any two chambers are the ends of a gallery. The distance between two chambers C and C' is the length $d(C, C')$ of a minimal gallery with ends C and C'.

A sub-building of a building A is a subset D of A such that D together with $\mathscr{C} \cap \mathfrak{P}(D)$ is a building.

a) Show that if A is a building, a facet has the same codimension in every chamber containing it; this allows us to speak of the codimension of a facet and of a panel of A without reference to a particular chamber. A morphism of a building B to A is a map $f$ from B to A such that the restriction of $f$ to every chamber C of B is a bijection from C to a chamber $f(C)$ of A. Show that the image under $f$ of a facet of B is a facet of the same codimension.

b) A building is called an apartment if every panel is contained in exactly two chambers. Show that if A is an apartment, every automorphism of A (i.e. every permutation of A preserving $\mathscr{C}$) that leaves fixed all the points of a chamber is the identity. More generally, let $\varphi$ be an endomorphism of A and let C be a chamber of A such that $\varphi(a) = a$ for all $a \in C$. Let $(C, C_1, \ldots, C_n)$ be a gallery of A. Show that, either the gallery $(C, \varphi(C_1), \ldots, \varphi(C_n))$ is not injective, or $\varphi(a) = a$ for all $a$ belonging to the union of the $C_i$.