Exercise 19, § 7

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

(a) If $g \in G$, every sequence $(s_1, \ldots, s_n)$ of elements of S such that $g = s_1 \ldots s_n$ and $s_i s_{i+1} \neq e$ for $1 \leq i < n$ is called a reduced decomposition of g. Show that every element of G admits one and only one reduced decomposition; the integer n is the length of g (cf. Exercise 26).

(b) Let Y be a subset of G containing e. Show the equivalence of the following conditions: $(b_1)$ Y is S-connected (cf. Exercise 18). $(b_2)$ For all $g \in Y$, if $(s_1, \ldots, s_n)$ is the reduced decomposition of g, then $s_1 \ldots s_i \in Y$ for $1 \leq i \leq n$. (c) Let $Y_1$ and $Y_2$ be two non-empty S-connected subsets of G such that $Y_1 \cap Y_2 = \varnothing$. Show that there exists at most one ordered pair

such that $y_1$ and $y_2$ are S-neighbours. There exists one if and only if $Y_1 \cup Y_2$ is S-connected; $Y_1$ and $Y_2$ are then said to be neighbours.

(d) Let $Y_1, \ldots, Y_n$ be a sequence of disjoint non-empty S-connected subsets of G. Suppose that $Y_i$ and $Y_{i+1}$ are neighbours for $1 \leq i < n$. Show that, if $n \geq 3$, $Y_1$ and $Y_n$ are not neighbours.