Exercise 38, § 7

Let $X$ be a finite set, $F(X)$ the free group constructed on $X$ and $S = X \cup X^{-1}$.

(a) Let $S_n$ be the set of products $s_1 \ldots s_m, s_i \in S, m \leq n$. Show that every connected $S$-component (cf. Exercise 18) of $F(X) - S_n$ contains one and only one element of the form $s_1 \ldots s_{n+1}$ with $s_i \in S$ and $s_i s_{i+1} \neq e$ for $1 \leq i \leq n$.

(b) Deduce from (a) a bijection of the set $B$ of ends of $F(X)$ (cf. Exercise 37) onto the set of infinite sequences $(s_1, \ldots, s_n, \ldots)$ of elements of $S$ such that $s_i s_{i+1} \neq e$ for all $i$. In particular, the number of ends of $F(X)$ is equal to $2^{k_0}$ if $\mathrm{Card}(X) \geq 2$.