Growth of quotients of groups acting by isometries on Gromov hyperbolic spaces

TL;DR

This paper proves that non-elementary groups acting properly and cocompactly on Gromov hyperbolic spaces exhibit growth tightness, meaning their exponential growth rate exceeds that of any infinite normal subgroup quotient, generalizing previous results.

Contribution

It establishes the growth tightness property for a broad class of groups acting on Gromov hyperbolic spaces, extending prior specific cases and answering an open question.

Findings

Groups acting properly and cocompactly on Gromov hyperbolic spaces are growth tight.

The exponential growth rate of such groups exceeds that of any quotient by an infinite normal subgroup.

The result unifies and generalizes previous work by Arzhantseva-Lysenok and Sambusetti.

Abstract

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric (pseudo)-distance induced by $X$ is greater than the exponential growth rate of any of its quotients by an infinite normal subgroup. This result generalizes from a unified framework previous works of Arzhantseva-Lysenok and Sambusetti, and provides an answer to a question of the latter.