Hierarchical hyperbolicity of graph products

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. Contents 1. Introduction 1 2. Background 5 3. The proto-hierarchy structure on a graph product 13 4. Graph products are relative HHGs 25 5. Some applications of hierarchical hyperbolicity 41 Appendix: Almost HHSs are HHSs 52 References 61

Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity

“…There are also various ways to combine HHSs and HHGs to produce new ones. For example, any direct product, or more generally graph product, of HHGs is an HHG [BR20]; many graphs of groups are HHGs [BHS19, BR18, RS20]; and both classes are closed under relative hyperbolicity [BHS19].…”

Extensions of multicurve stabilizers are hierarchically hyperbolic

“…Hierarchical hyperbolicity axiomatizes this theory, describing a class of spaces whose coarse geometry is encoded in a collection of projections onto hyperbolic metric spaces that are organized by a set of combinatorial relations. Remarkably, the class of hierarchically hyperbolic spaces encompasses a variety of groups beyond the mapping class group including the fundamental group of most 3-manifolds [BHS19], many cocompactly cubulated groups [BHS17b,HS20], Artin groups of extra large type [HMS], and several combinations of hyperbolic groups [BR20a,RS,BR20b]. Hierarchical hyperbolicity also describes the coarse geometry of a number of other groups and spaces associated to surfaces such as Teichmüller space with both the Teichmüller and Weil-Peterson metrics [BHS17b, MM99, MM00, BKMM12, Bro03, Dur16, Raf07, EMR17], the genus 2 handlebody group [Che20], the π 1 pSq-extensions of lattice Veech groups [DDLS], certain quotients of the mapping class group [BHS17a,BHMS20], and a wide variety of graphs built from curves on surfaces [Vok17].…”