Groups acting on CAT(0) cube complexes with uniform exponential growth

TL;DR

This paper investigates the conditions under which groups acting on CAT(0) cube complexes exhibit uniform exponential growth, extending previous results to broader classes of groups and actions.

Contribution

It generalizes existing work by showing uniform exponential growth for groups acting without fixed points on CAT(0) square complexes and for certain improperly acting groups.

Findings

Groups acting without fixed points on CAT(0) square complexes have uniform exponential growth or stabilize a Euclidean subcomplex.

Finitely generated subgroups of the Higman group and triangle-free Artin groups have uniform exponential growth.

Non-virtually abelian groups acting freely on CAT(0) cube complexes with isolated flats have uniform exponential growth.

Abstract

We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev that considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that non-virtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth.