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

Abstract

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 non-virtually 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.

arXiv commentMinor changes to address referee comments. Final version to appear in Algebraic & Geometric Topology

References in corpus (12)

• Bounded cohomology of subgroups of mapping class groups
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Simplicite de groupes d'automorphismes d'espaces a courbure negative
• Hadamard spaces with isolated flats
• A random tunnel number one 3-manifold does not fiber over the circle
• On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
• Quasi-isometric classification of right-angled Artin groups I: the finite out case
• On the cubical geometry of Higman's group
• Uniform exponential growth for CAT(0) square complexes
• Large facing tuples and a strengthened sector lemma
• Ping-pong in Hadamard manifolds
• Lower bounds on cubical dimension of $C^{\prime }(1/6)$ groups

Cited by in corpus (2)

• Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
• Extensions of hyperbolic groups have locally uniform exponential growth