Uniform exponential growth for CAT(0) square complexes

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, 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.

Lower bounds on cubical dimensionof $C’(1/6)$ groups

“…Remark 4.1. The case n = 2 of Theorem 1.2 can be deduced from the work of Kar and Sageev who study uniform exponential growth of groups acting freely on CAT(0) square complexes [KS16]. They prove that for any two elements x, y there exists a pair of words of length at most 10 in x, y that freely generates a free semigroup, unless x, y is virtually abelian.…”

“…Pride's construction has been revisited in [JW17]. We observe that the case n = 2 can be deduced from the work of Kar and Sageev who study uniform exponential growth of groups acting freely on CAT(0) square complexes [KS16]. See Remark 4.1.…”

“…I would like to thank my supervisors Piotr Przytycki and Daniel Wise. I would also like to thank Carolyn Abbot, Yen Duong, Teddy Einstein, Justin Lanier, Thomas Ng and Radhika Gupta for helpful discussion on [KS16]. The author was partially supported by (Polish) Narodowe Centrum Nauki, grant no.…”

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

“…Kar and Sageev showed that if a group acts freely on a CAT(0) square complex, then either it has uniform exponential growth or it is virtually abelian [KS19] . In this article, we generalize the result of Kar and Sageev by removing the assumption of a free action.…”

Large facing tuples and a strengthened sector lemma

“…Find a constant λ such that either G is virtually abelian or, for any finite generating set of G, the λ-ball in the corresponding Cayley graph of G contains two elements that freely generate a free (semi)group. There is quite a strong result about this in the 2-dimensional case, due to Kar and Sageev [KS19], and this is an actively-studied question in higher dimensions.…”