Exercise 39, § 7

\* Let G be a finitely generated group, and let $\mathscr{F}$ be the set of finite symmetric subsets of G, containing e, which generate G. (a) If $X \in \mathscr{F}$ and $n \in \mathbf{N}$, denote by $d_n(X)$ the number of elements of G of the form $x_1 \ldots x_n$, with $x_i \in X$. If $X \in \mathscr{F}$, $Y \in \mathscr{F}$, now prove that there exists an integer $a \geqslant 1$ 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 if G is infinite, now prove that $d_n(X) < d_{n+1}(X)$ for all n. Deduce that $e(G)$ is $\geqslant 1$. (c) Let H be a finitely generated group. Now prove that $e(H) \leqslant e(G)$ if H is isomorphic to a subgroup (resp. to a quotient group) of G. If E is an extension of G by H, now prove that $e(E) \geqslant e(H) + e(G)$, with equality when $E = G \times H$. (d) Now prove 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 $d_n(X) \geqslant a^n$ for all sufficiently large n. Now prove that, if property (i) is verified by an element of $\mathscr{F}$, it is by every element of $\mathscr{F}$. G is then said to be of exponential type; this entails $e(G) = +\infty$. (f) Let $G_1$ and $G_2$ be two groups, containing the same subgroup A, and let $H = G_1 *_A G_2$ be the corresponding amalgamated sum. Suppose that $G_1$ and $G_2$ are finitely generated (and hence H also), and that $G_i \neq A$ for $i = 1, 2$. Now prove that H is of exponential type if at least one of the indices $(G_1 : A)$, $(G_2 : A)$ is $\geqslant 3$. (g) Now prove that a free group of rank $\geqslant 2$ is of exponential type.*