Uniform exponential growth for CAT(0) square complexes

Kar, Aditi; Sageev, Michah

doi:10.2140/agt.2019.19.1229

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Uniform exponential growth for CAT(0) square complexes

Aditi Kar and Michah Sageev

Algebraic & Geometric Topology 19 (2019) 1229–1245

DOI: 10.2140/agt.2019.19.1229

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 $10$ 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.

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Keywords

uniform exponential growth, CAT(0) cubical groups

Mathematical Subject Classification 2010

Primary: 20F65

References

Publication

Received: 21 August 2017

Revised: 18 June 2018

Accepted: 5 November 2018

Published: 21 May 2019

Authors

Aditi Kar
Department of Mathematics Royal Holloway, University of London Egham United Kingdom

Michah Sageev
Department of Mathematics Israel Institute of Technology Haifa Israel

Volume 19, issue 3 (2019)