  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.

Group Theory;Geometric Topology