Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

Jason Behrstock, Mark Hagen and Alessandro Sisto

Vol. 299 (2019), No. 2, 257–338

DOI: 10.2140/pjm.2019.299.257

Abstract

We introduce a number of tools for finding and studying hierarchically hyperbolic spaces (HHS), a rich class of spaces including mapping class groups of surfaces, Teichmüller space with either the Teichmüller or Weil–Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHS satisfy a Masur–Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur–Minsky hierarchy machinery. We then study examples of HHS; for instance, we prove that when $M$ is a closed irreducible $3$-manifold then ${\pi }_{1}M$ is an HHS if and only if it is neither $\mathit{Nil}$ nor $\mathit{Sol}$. We establish this by proving a general combination theorem for trees of HHS (and graphs of HH groups). We also introduce a notion of “hierarchical quasiconvexity”, which in the study of HHS is analogous to the role played by quasiconvexity in the study of Gromov-hyperbolic spaces.

Keywords

geometric group theory, hierarchically hyperbolic, mapping class group

Mathematical Subject Classification 2010

Primary: 20F36, 20F65, 20F67

Milestones

Received: 25 January 2018

Revised: 15 November 2018

Accepted: 27 November 2018

Published: 21 May 2019