Exercise 13, § 1

Let $(W, S)$ be a Coxeter system. For $s, s' \in S$, define the sequence $\mathbf{a}(s, s')$ by means of the following rules:

(i) if $ss'$ is of infinite order, $\mathbf{a}(s, s')$ is the empty sequence; (ii) if $ss'$ is of finite order $m$ the sequence $\mathbf{a}(s, s')$ is of length $m$ and its even (resp. odd) numbered terms are equal to $s'$ (resp. $s$).

Denote by $a(s, s')$ the product of the sequence $\mathbf{a}(s, s')$.

a) Show that the generating set $S$ and the relations $s^2 = 1$ and $\mathbf{a}(s, s') = \mathbf{a}(s', s)$ form a presentation of the group $W$.

b) Let $q$ be an integer $\geqslant 1$. Let $\Sigma_q$ be the set of sequences of $q$ elements of $S$ and let $R_q$ be the smallest equivalence relation on $\Sigma_q$ for which sequences of the form $(A, \mathbf{a}(s, s'), B)$ and $(A, \mathbf{a}(s', s), B)$ (where $s, s' \in S$ and $A$ and $B$ are sequences of elements of $S$) are equivalent. Let $\Sigma_q^r$ be the set of sequences $s = (s_1, \ldots, s_q)$ such that $w(s) = s_1 \ldots s_q$ is of length $q$. Show that sequences $s, s' \in \Sigma_q^r$ are equivalent modulo $R_q$ if and only if $w(s) = w(s')$ (argue by induction on $q$ and apply Prop. 5 of no. 5).

c) Show that a sequence $s \in \Sigma_q$ does not belong to $\Sigma_q^r$ if and only if $s$ is equivalent modulo $R_q$ to a sequence in which two consecutive terms are equal. (Argue by induction on $q$ and reduce to the case of a sequence $(s_1, \ldots, s_q)$ which does not belong to $\Sigma_q^r$ but which is such that $(s_1, \ldots, s_{q-1})$ and $(s_2, \ldots, s_q)$ belong to $\Sigma_{q-1}^r$; use Exerc. 1 to show that $s_1 \ldots s_{q-1} = s_2 \ldots s_q$ and apply b).)