# A note on growth of hyperbolic groups

## TL;DR

This paper provides an alternative proof that the growth function of a non-elementary hyperbolic group is exponential, with explicit bounds, confirming the exponential growth rate of such groups.

## Contribution

It offers a new proof of the exponential growth rate of hyperbolic groups, establishing explicit bounds for the growth function.

## Key findings

- Growth of hyperbolic groups is exponential.
- Explicit constants for growth bounds are provided.
- Alternative proof method for known growth properties.

## 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 $\Gamma(G,X)$.