Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups

TL;DR

This paper proves that all hierarchically hyperbolic spaces have finite asymptotic dimension, improves bounds on the mapping class group's dimension, and introduces a small-cancellation theory for hierarchically hyperbolic groups.

Contribution

It establishes finite asymptotic dimension for hierarchically hyperbolic spaces and groups, and develops a small-cancellation framework for these groups, expanding the class of known examples.

Findings

Finite asymptotic dimension for all hierarchically hyperbolic spaces.

Quadratic upper bound on the asymptotic dimension of the mapping class group.

New small-cancellation results for hierarchically hyperbolic groups.

Abstract

We prove that all hierarchically hyperbolic spaces have finite asymptotic dimension and obtain strong bounds on these dimensions. One application of this result is to obtain the sharpest known bound on the asymptotic dimension of the mapping class group of a finite type surface: improving the bound from exponential to at most quadratic in the complexity of the surface. We also apply the main result to various other hierarchically hyperbolic groups and spaces. We also prove a small-cancellation result namely: if $G$ is a hierarchically hyperbolic group, $H\leq G$ is a suitable hyperbolically embedded subgroup, and $N\triangleleft H$ is "sufficiently deep" in $H$, then $G/\langle\langle N\rangle\rangle$ is a relatively hierarchically hyperbolic group. This new class provides many new examples to which our asymptotic dimension bounds apply. Along the way, we prove new results about the structure of HHSs, for example: the associated hyperbolic spaces are always obtained, up to quasi-isometry, by coning off canonical coarse product regions in the original space (generalizing a relation established by Masur--Minsky between the complex of curves of a surface and Teichm\"{u}ller space).