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

We prove that all hierarchically hyperbolic groups have finite asymptotic dimension. 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≤G is a suitable hyperbolically embedded subgroup, and N◃H is 'sufficiently deep' in H, then G/N 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 and Minsky between the complex of curves of a surface and Teichmüller space).