Vol. 299, No. 2, 2019

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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

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 π1M is an HHS if and only if it is neither Nil nor 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.

geometric group theory, hierarchically hyperbolic, mapping class group
Mathematical Subject Classification 2010
Primary: 20F36, 20F65, 20F67
Received: 25 January 2018
Revised: 15 November 2018
Accepted: 27 November 2018
Published: 21 May 2019
Jason Behrstock
The Graduate Center and Lehman College
New York, NY
United States
Mark Hagen
School of Mathematics
University of Bristol
United Kingdom
Alessandro Sisto
Departement Mathematik
ETH Zürich