A note on growth of hyperbolic groups

Abstract

The following short note provides an alternative proof of a result of Coornaert: namely, that given a non-elementary word-hyperbolic group $G$ with a finite generating set $X$, there exist constants $\lambda,D > 1$ such that \[ D^{-1}\lambda^n \leq |B_{G,X}(n)| \leq D \lambda^n \] for all $n \geq 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $\Gamma(G,X)$.