Hierarchically hyperbolic spaces II: Combination theorems and the distance formula

Jason Behrstock, Mark Hagen, Alessandro Sisto

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

8 Citations (Scopus)
106 Downloads (Pure)

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 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.
Original languageEnglish
Pages (from-to)257-338
Number of pages82
JournalPacific Journal of Mathematics
Volume299
Issue number2
Early online date21 May 2019
DOIs
Publication statusPublished - 2019

Keywords

  • geometric group theory
  • hierarchically hyperbolic
  • mapping class group

Fingerprint Dive into the research topics of 'Hierarchically hyperbolic spaces II: Combination theorems and the distance formula'. Together they form a unique fingerprint.

Cite this