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Hierarchically hyperbolic spaces. II: Combination theorems and the distance formula. (English) Zbl 1515.20208

Summary: 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 \(\pi_1M\) is an HHS if and only if it is neither \(\mathrm{Nil}\) nor \(\mathrm{Sol}\). 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.
For Part I see [the authors, Geom. Topol. 21, No. 3, 1731–1804 (2017; Zbl 1439.20043)].

MSC:

20F65 Geometric group theory
20F36 Braid groups; Artin groups
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory

Citations:

Zbl 1439.20043
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References:

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