Mathematics > Group Theory
[Submitted on 1 Sep 2019 (v1), last revised 4 Nov 2021 (this version, v2)]
Title:Hierarchically hyperbolic groups and uniform exponential growth
View PDFAbstract: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.
Submission history
From: Thomas Ng [view email][v1] Sun, 1 Sep 2019 17:37:59 UTC (44 KB)
[v2] Thu, 4 Nov 2021 15:27:05 UTC (89 KB)
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