Product set growth in groups and hyperbolic geometry

TL;DR

This paper establishes growth bounds for finite subsets in hyperbolic groups and related structures, generalizing previous results and answering a specific open question in geometric group theory.

Contribution

It proves new lower bounds on the growth of finite subsets in hyperbolic groups and groups acting on hyperbolic spaces, extending prior work and resolving an open problem.

Findings

Growth bounds for hyperbolic groups

Extension to groups acting on hyperbolic spaces

Answer to Button's open question

Abstract

Generalising results of Razborov and Safin, and answering a question of Button, we prove that for every hyperbolic group there exists a constant $\alpha >0$ such that for every finite subset $U$ that is not contained in a virtually cyclic subgroup $|U^n|\geqslant (\alpha |U|)^{[(n+1)/2]}$. Similar estimates are established for groups acting acylindrically on trees or hyperbolic spaces.