Exercise 32, § 7

Let A and B be two groups and $G = A * B$ their free product. Let X be the set of commutators $(a, b)$ with $a \in A - \{e\}$ and $b \in B - \{e\}$ and $S = X \cup X^{-1}$.

(a) Show that $X \cap X^{-1} = \varnothing$ and that, if $x$ is an element of X, there exists a unique ordered pair $a, b$ in $(A - \{e\}) \times (B - \{e\})$ such that $(a, b) = x$.

(b) Let $n \geq 1$ and let $s_1, \ldots, s_n$ be a sequence of elements of S such that $s_i s_{i+1} \neq e$ for $1 \leq i < n$; let $a_n \in A - \{e\}, b_n \in B - \{e\}, \varepsilon(n) = \pm 1$ be such that $s_n = (a_n, b_n)^{\varepsilon(n)}$. Let $g = s_1 \ldots s_n$. Show, by induction on $n$, that the reduced decomposition of $g$ is of length $\geq n + 3$ and terminates either with $a_n b_n$ if $\varepsilon(n) = 1$ or with $b_n a_n$ if $\varepsilon(n) = -1$.

(c) Let R be the kernel of the canonical projection $A * B \to A \times B$ corresponding (no. 3) to the canonical injections $A \to A \times B$ and $B \to A \times B$. Show that R is a free group with basic family X. (Use (b) to prove that X is free; if $R_X$ is the subgroup of G generated by X, show that $R_X$ is normal; deduce that it coincides with R.)