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 $λ,D > 1$ such that \[ D^{-1}λ^n \leq |B_{G,X}(n)| \leq D λ^n \] for all $n \geq 0$, where $B_{G,X}(n)$ is the ball of radius $n$ in the Cayley graph $Γ(G,X)$.