Algebraic & Geometric Topology

Uniform exponential growth for CAT(0) square complexes

Aditi Kar and Michah Sageev

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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

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

Received: 21 August 2017
Revised: 18 June 2018
Accepted: 5 November 2018
First available in Project Euclid: 29 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

uniform exponential growth CAT(0) cubical groups


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.

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