Exercise 19, § 4

Let (W, S) be the Coxeter system corresponding to the graph

!A Coxeter graph

This is a system of compact hyperbolic type (cf. Exerc. 15). The coefficients of the form $B_M$ with respect to the basis $(e_s)$ belong to the subring A of the field $\mathbf{Q}(\sqrt{5})$ consisting of the elements of this field that are integers over $\mathbf{Z}$.

a) Let $\sigma$ be the embedding of K into $\mathbf{R}$ that maps $\sqrt{5}$ to $-\sqrt{5}$. Show that the transform $\sigma(B_M)$ of the form $B_M$ by $\sigma$ is non-degenerate and positive.

b) Let G be the orthogonal group of $B_M$, and let $G_\sigma$ be that of $\sigma(B_M)$. Let $G_A$ be the subgroup of G consisting of the elements whose matrices with respect to $(e_s)$ have coefficients in A. Show that $G_A$ can be identified with a discrete subgroup of $G \times G_\sigma$ and then, using a), show that $G_A$ is a discrete subgroup of G.

c) Show that W is a subgroup of G_A of finite index.

d) Prove analogous results for the other graphs in Exerc. 15 (the field Q(\sqrt{5}) sometimes being replaced by Q(\sqrt{2}), with the exception of the graph

!Graph with two nodes labeled 4

treated in Exerc. 18, and of the graph⁸

!Graph with two nodes labeled 4 and one node labeled 5