Growth of periodic quotients of hyperbolic groups

TL;DR

This paper investigates how the exponential growth rate of periodic quotients of non-elementary torsion-free hyperbolic groups approaches that of the original group as the exponent n increases, providing convergence estimates.

Contribution

It proves the convergence of growth rates of periodic quotients to that of the original hyperbolic group and offers explicit estimates for the convergence rate.

Findings

Growth rate of G/G^n approaches that of G as n increases.

Convergence rate estimates are provided.

Results apply to non-elementary torsion-free hyperbolic groups.

Abstract

Let G be a non-elementary torsion-free hyperbolic group. We prove that the exponential growth rate of the periodic quotient G/G^n tends to the one of G as n odd approaches infinity. Moreover we provide an estimate at which the convergence is taking place.