Uniform exponential growth for CAT(0) square complexes

TL;DR

This paper investigates uniform exponential growth in groups acting on CAT(0) square complexes, establishing conditions under which free semigroups exist or elements stabilize flats, with implications for growth bounds.

Contribution

It proves a dichotomy for hyperbolic automorphisms on CAT(0) square complexes, providing a uniform lower bound for growth constants across a broad class of groups.

Findings

Existence of free semigroups under certain conditions

All elements stabilize flats or generate free semigroups

Uniform lower bound for growth constant is established

Abstract

In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if $F$ is a finite collection of hyperbolic automorphisms of a CAT(0) square complex $X$, then either there exists a pair of words of length at most 10 in $F$ which freely generate a free semigroup, or all elements of $F$ stabilize a flat (of dimension 1 or 2 in $X$). As a corollary, we obtain a lower bound for the growth constant, $\sqrt[10]{2}$, which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.