Uniform exponential growth for groups with proper product actions on hyperbolic spaces

TL;DR

This paper proves that groups acting properly on products of hyperbolic spaces exhibit uniform exponential and product set growth under certain conditions, extending to hierarchically hyperbolic groups and various geometric group classes.

Contribution

It establishes uniform exponential and product set growth for groups with proper actions on hyperbolic space products, including new classes like hierarchically hyperbolic groups.

Findings

Finitely generated non-virtually abelian subgroups have uniform exponential growth.

Groups with weak acylindrical actions have uniform product set growth, with two exceptions.

Complete classification of subgroups with product set growth in various geometric contexts.

Abstract

This paper studies the locally uniform exponential growth and product set growth for a finitely generated group $G$ acting properly on a finite product of hyperbolic spaces. Under the assumption of coarsely dense orbits or shadowing property on factors, we prove that any finitely generated non-virtually abelian subgroup has uniform exponential growth. These assumptions are fulfilled in many hierarchically hyperbolic groups, including mapping class groups, specially cubulated groups and BMW groups. Moreover, if $G$ acts weakly acylindrically on each factor, we show that, with two exceptional classes of subgroups, $G$ has uniform product set growth. As corollaries, this gives a complete classification of subgroups with product set growth for any group acting discretely on a simply connected manifold with pinched negative curvature, for groups acting acylindrically on trees, and for 3-manifold groups.