Hierarchically hyperbolic groups and uniform exponential growth

TL;DR

This paper establishes conditions under which hierarchically hyperbolic groups exhibit uniform exponential growth, providing new insights and proofs for groups with non-positive curvature properties, including some acting on CAT(0) cubical spaces.

Contribution

It offers the first proof of uniform exponential growth for certain groups acting on high-dimensional CAT(0) cubical spaces and provides a quasi-isometric characterization of groups lacking this growth.

Findings

Hierarchically hyperbolic groups with acylindrical hyperbolicity have uniform exponential growth.

A quasi-isometric criterion characterizes groups without uniform exponential growth.

New proofs of exponential growth for groups with non-positive curvature properties.

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.