Hierarchical hyperbolicity of graph products

TL;DR

This paper proves that graph products of finitely generated groups are hierarchically hyperbolic relative to their vertex groups, extending the class of known hierarchically hyperbolic groups and spaces with new geometric insights.

Contribution

It establishes the hierarchical hyperbolicity of graph products and their syllable metrics, and shows that graph products of hierarchically hyperbolic groups are also hierarchically hyperbolic, removing previous restrictions.

Findings

Graph products are hierarchically hyperbolic relative to vertex groups.

Syllable metric on graph products forms a hierarchically hyperbolic space.

Graph products of hierarchically hyperbolic groups are hierarchically hyperbolic groups.

Abstract

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.