Hyperbolic HHS II: Graphs of hierarchically hyperbolic groups

TL;DR

This paper demonstrates that the fundamental groups of graphs of hierarchically hyperbolic groups (HHG) can themselves be endowed with HHG structures, advancing understanding of hierarchical hyperbolicity in complex group constructions.

Contribution

It introduces a method to incorporate HHG structures of embedded subgroups into larger groups, generalizing previous procedures and expanding the applicability of hierarchical hyperbolicity.

Findings

Fundamental groups of graphs of HHGs admit HHG structures.

Hierarchical quasi convexity coincides with quasi-convexity in hyperbolic HHS.

A new technique to embed HHG structures of subgroups into ambient groups.

Abstract

In this paper we consider a large family of graphs of hierarchically hyperbolic groups (HHG) and show that their fundamental groups admit HHG structures. To do that, we will investigate the notion of hierarchical quasi convexity and show that for a hyperbolic HHS it coincides with the notion of quasi-convexity. The main technical result, for which we expect further applications, is that it is possible to incorporate the HHG structure of a hierarchically hyperbolically embedded subgroup into the HHG structure of the ambient group. This generalizes and provides some additional details to a procedure described in [DHS17].