14.7 (1999)

Let $\Gamma_g$ be the fundamental group of a closed surface of genus $g \geqslant 2$. For each finite system $S$ of generators of $\Gamma_g$, let $\beta_S(n)$ denote the number of elements of $\Gamma_g$ which can be written as products of at most $n$ elements of $S \cup S^{-1}$, and let $\omega(\Gamma_g, S) = \limsup_{n \to \infty} \sqrt[n]{\beta_S(n)}$ be the growth rate of the sequence $(\beta_S(n))_{n\geqslant 0}$. Compute the infimum $\omega(\Gamma_g)$ of the $\omega(\Gamma_g, S)$ over all finite sets of generators of $\Gamma_g$.

It is easy to see that, for a free group $F_k$ of rank $k \geqslant 2$, the corresponding infimum is $\omega(F_k) = 2k - 1$ (M. Gromov, Structures métriques pour les variétés riemanniennes, Cedic/F. Nathan, Paris, 1981, Ex. 5.13). As any generating set of $\Gamma_g$ contains a subset of $2g - 1$ elements generating a subgroup of infinite index in $\Gamma_g$ with abelianization $\mathbb{Z}^{2g-1}$, hence a subgroup which is free of rank $2g - 1$, it follows that $\omega(\Gamma_g) \geqslant 4g - 3$.