Exercise 39, § 7

*Let $G$ be a finitely generated group and $\mathscr{F}$ the set of symmetric finite subsets of $G$ containing $e$, which generate $G$.

(a) If $X \in \mathscr{F}$ and $n \in \mathbf{N}$, let $d_n(X)$ denote the number of elements in $G$ of the form $x_1 \ldots x_n$ with $x_i \in X$. If $X, Y \in \mathscr{F}$, show that there exists an integer $a \geq 1$ such that

Deduce that $\limsup_{n \to \infty} \frac{\log(d_n(X))}{\log(n)}$ does not depend on the choice of $X$ in $\mathscr{F}$; this limit (finite or infinite) is denoted by $e(G)$.

(b) If $X \in \mathscr{F}$ and $G$ is infinite, show that $d_n(X) < d_{n+1}(X)$ for all $n$. Deduce that $e(G)$ is $\geq 1$.

(c) Let $H$ be a finitely generated group. Show that $e(H) \leq e(G)$ if $H$ is isomorphic to a subgroup (resp. a quotient group) of $G$. If $E$ is an extension of $G$ by $H$, show that $e(E) \geq e(H) + e(G)$, with equality when $E = G \times H$.

(d) Show that $e(\mathbf{Z}^n) = n$.

(e) Let $X \in \mathscr{F}$. Consider the following property: (i) There exists a real number $a > 1$ such that $d_n(X) \geq a^n$ for all sufficiently large $n$. Show that, if property (i) holds for one element of $\mathscr{F}$, it holds for every element of $\mathscr{F}$. $G$ is then said to be of exponential type; this implies $e(G) = +\infty$.

(f) Let $G_1$ and $G_2$ be two groups containing the same subgroup $A$ and let $H = G_1 *_A G_2$ the corresponding amalgamated sum. Suppose $G_1$ and $G_2$ are finitely generated (hence also $H$ is) and $G_i \neq A$ for $i = 1, 2$. Show that $H$ is of exponential type if at least one of the indices $(G_1 : A), (G_2 : A)$ is $\geq 3$.

(g) Show that a free group of rank $\geq 2$ is of exponential type.*