Exercise 18, § 4

Assume that (W, S) is of hyperbolic type, and that W leaves stable a lattice $\Gamma$ of E. Let G be the orthogonal group of $B_M$, and let $G(\Gamma)$ be the subgroup of elements $g \in G$ such that $g\Gamma = \Gamma$. Show that $G(\Gamma)$ is a discrete subgroup of G. Show that W is a subgroup of finite index of $G(\Gamma)$ (use the fact that the measure of $W \setminus G$ is finite).

*If (W, S) is actually of compact hyperbolic type, show that the corresponding Coxeter graph is isomorphic to one of the following (use Exercs. 4, 6 and 15):

!Five Coxeter graphs

and

Show that all the groups corresponding to the graphs of Exerc. 16 (with the exception of the last four) leave stable a lattice (use the method of Exerc. 6).*