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

Abstract

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). Contents

Coarse medians and Property A

“…While this property is stronger (see [BHS15b,Section 7]), and somewhat more involved than coarse medians it is also substantially more powerful: it implies even finite asymptotic dimension [BHS15a]. Having finite asymptotic dimension is a strictly stronger property than Property A.…”

A characterization for asymptotic dimension growth

“…Hierarchically hyperbolic spaces are examples of coarse median spaces, see [2], hence our theorem is broader in scope, though with a weaker conclusion, than the finite asymptotic dimension result proven in [1]. We expect the following general result.…”

“…Among those are, for instance, finitely generated abelian groups, free groups of finite rank, Gromov hyperbolic groups [14,24], mapping class groups [5], CAT(0) cube complexes of finite dimension [29], see [3] for an excellent survey of these and other results. Recently Behrstock, Hagen and Sisto introduced the powerful new notion of hierarchically hyperbolic spaces and showed that these have finite asymptotic dimension [1], recovering a number of the above results, including notably mapping class groups and a number of CAT(0) cube complexes.…”

“…There is also another metric associated with X. Let X (1) be the 1-skeleton of X, that is a graph with the vertex set V = X (0) . We equip V with the edge-path metric d, which is the minimal number of edges in a path connecting two given vertices.…”

“…We equip V with the edge-path metric d, which is the minimal number of edges in a path connecting two given vertices. Clearly, when X (1)…”