Exercise 14, § 1

Let $(W, S)$ be a Coxeter system and let $(\Gamma, f)$ be its Coxeter graph. Let $k$ be an integer $\geqslant 3$. If $a$ is an edge of $\Gamma$, put $f_k(a) = f(a)$ if $f(a) \neq \infty$ and $f_k(a) = k$ if $f(a) = \infty$. Let $(W_k, S)$ be a Coxeter system whose Coxeter graph is equal to $(\Gamma, f_k)$ (Chap. V, § 4, no. 3, Cor. of Prop. 4). Show that there exists a unique homomorphism $\varphi_k$ from $W$ to $W_k$ inducing the identity on $S$. Show that if $k$ divides $k'$, there exists a unique homomorphism $\varphi_{k,k'}$ from $W_{k'}$ to $W_k$ such that $\varphi_k = \varphi_{k,k'} \circ \varphi_{k'}$. Show that the homomorphism $(\varphi_k)$ from $W$ to the projective limit of the $W_k$ is injective (use Exerc. 13) c)), but in general not surjective (for example, in the case of the infinite dihedral group).*