## Algebraic & Geometric Topology

### Uniform exponential growth for CAT(0) square complexes

#### 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 $X$, then either there exists a pair of words of length at most $1 0$ 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, $2 1 0$, 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 3 (2019), 1229-1245.

Dates
Revised: 18 June 2018
Accepted: 5 November 2018
First available in Project Euclid: 29 May 2019

https://projecteuclid.org/euclid.agt/1559095424

Digital Object Identifier
doi:10.2140/agt.2019.19.1229

Mathematical Reviews number (MathSciNet)
MR3954280

Zentralblatt MATH identifier
07142601

Subjects

#### Citation

Kar, Aditi; Sageev, Michah. Uniform exponential growth for CAT(0) square complexes. Algebr. Geom. Topol. 19 (2019), no. 3, 1229--1245. doi:10.2140/agt.2019.19.1229. https://projecteuclid.org/euclid.agt/1559095424

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