Extensions of hyperbolic groups have locally uniform exponential growth

TL;DR

This paper develops a framework to analyze when extensions of hyperbolic and right-angled Artin groups preserve locally uniform exponential growth, expanding the class of groups known to have this property.

Contribution

It introduces a quantitative subgroup alternative framework and applies it to show that various extended groups, including automorphism groups, have locally uniform exponential growth.

Findings

Extensions of hyperbolic groups retain exponential growth properties.

Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.

The subgroup alternative framework applies to a broad class of groups.

Abstract

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that automorphism groups of torsion-free one-ended toral relatively hyperbolic groups and certain right-angled Artin groups satisfy our quantitative subgroup alternative.