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 refined combination theorem for hierarchically hyperbolic groups
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abstract
In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering a question posed by Behrstock, Hagen, and Sisto. In order to operate in such a general setting, we establish a number of structural results for hierarchically hyperbolic spaces and hieromorphisms (that is, morphisms between such spaces), and we introduce two new notions for hierarchical hyperbolicity, that is concreteness and the intersection property, proving that they are satisfied in all known examples.
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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.