A virtually torsion-free hierarchically hyperbolic group either has uniform exponential growth or its Cayley graph is quasi-isometric to Z times a space.
A Wilson group of non-uniformly exponential growth
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Hierarchically hyperbolic groups and uniform exponential growth
A virtually torsion-free hierarchically hyperbolic group either has uniform exponential growth or its Cayley graph is quasi-isometric to Z times a space.