A refined combination theorem for hierarchically hyperbolic groups

TL;DR

This paper extends the combination theorem for hierarchically hyperbolic groups, showing that finite graph products of such groups are also hierarchically hyperbolic, and introduces new structural notions for these spaces.

Contribution

It generalizes the combination theorem for hierarchically hyperbolic groups and introduces the concepts of concreteness and the intersection property.

Findings

Finite graph products of hierarchically hyperbolic groups are hierarchically hyperbolic.

Established structural results for hierarchically hyperbolic spaces and morphisms.

Introduced and validated the notions of concreteness and the intersection property.

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.