Hierarchically hyperbolic groups and uniform exponential growth

2019/09/01 by Carolyn Abbott, Abbott, Carolyn, Thomas Ng +7 · 2 citations
Mathematics · #20F65 #20F67 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1909.00439

openalex publication_date 2019/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterization of hierarchically hyperbolic groups without uniform exponential growth. To achieve this, we gain new insights on the structure of certain classes of hierarchically hyperbolic groups. Our methods give a new unified proof of uniform exponential growth for several examples of groups with notions of non-positive curvature. In particular, we obtain the first proof of uniform exponential growth for certain groups that act geometrically on CAT(0) cubical spaces of dimension 3 or more. Under additional hypotheses, we show that a quantitative Tits alternative holds for hierarchically hyperbolic groups.

Cited by

• The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces
• Embedability of right-angled Artin groups into hierarchically hyperbolic groups

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