#### Volume 19, issue 3 (2019)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Uniform exponential growth for CAT(0) square complexes

### Aditi Kar and Michah Sageev

Algebraic & Geometric Topology 19 (2019) 1229–1245
##### Abstract

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 $X\phantom{\rule{0.3em}{0ex}}$, 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, $\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sqrt[10]{2}$, 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.

##### Keywords
uniform exponential growth, CAT(0) cubical groups
Primary: 20F65