A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces

TL;DR

This paper establishes a correspondence between hierarchically hyperbolic spaces (HHSs) and combinatorial structures, enabling new constructions and clarifications of the HHS framework, with applications to groups like mapping class groups.

Contribution

It shows that HHSs satisfying natural conditions can be characterized by combinatorial structures, enhancing understanding and construction of HHSs.

Findings

HHSs admit a combinatorial structure under natural assumptions

Connections between HHS notions and lattice theory are uncovered

Clarifies application of the combinatorial HHS criterion to examples

Abstract

The combinatorial hierarchical hyperbolicity criterion is a very useful way of constructing new hierarchically hyperbolic spaces (HHSs). We show that, conversely, HHSs satisfying natural assumptions (satisfied, for example, by mapping class groups) admit a combinatorial HHS structure. This can be useful in constructions of new HHSs, and also our construction clarifies how to apply the combinatorial HHS criterion to suspected examples. We also uncover connections between HHS notions and lattice theory notions.