Hierarchically Hyperbolic Spaces Ii: Combination Theorems and the Distance Formula
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چکیده
We introduce a number of tools for nding 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 de ning an HHS. We prove that all HHSs 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 HHSs; 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 HHSs (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.
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تاریخ انتشار 2017