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

Jason Behrstock, Mark F. Hagen, Alessandro Sisto

Research output: Contribution to journalArticle (Academic Journal)peer-review

24 Citations (Scopus)
320 Downloads (Pure)


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).
Original languageEnglish
Pages (from-to)890-926
JournalProceedings of the London Mathematical Society
Issue number5
Publication statusPublished - 27 Feb 2017

Bibliographical note

Minor revisions in Section 6. This is the version accepted for publication


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