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

Radhika Gupta, Kasia Jankiewicz and Thomas Ng

Algebraic & Geometric Topology 23 (2023) 13–42

We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev that considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that nonvirtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth.

uniform exponential growth, CAT(0) cube complexes
Mathematical Subject Classification
Primary: 20E07, 20F65, 20F67
Secondary: 57M60
Received: 8 July 2020
Revised: 19 September 2021
Accepted: 24 October 2021
Published: 27 March 2023
Radhika Gupta
School of Mathematics
Tata Institute of Fundamental Research
Kasia Jankiewicz
Department of Mathematics
University of California, Santa Cruz
Santa Cruz, CA
United States
Thomas Ng
Mathematics Department
Technion - Israel Institute of Technology

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