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Uniform exponential growth for CAT(0) square complexes

Aditi Kar and Michah Sageev

Algebraic & Geometric Topology 19 (2019) 1229–1245

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, 210, 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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uniform exponential growth, CAT(0) cubical groups
Mathematical Subject Classification 2010
Primary: 20F65
Received: 21 August 2017
Revised: 18 June 2018
Accepted: 5 November 2018
Published: 21 May 2019
Aditi Kar
Department of Mathematics
Royal Holloway, University of London
United Kingdom
Michah Sageev
Department of Mathematics
Israel Institute of Technology