Hierarchically hyperbolic spaces II: Combination theorems and the distance formula
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2015-12-18
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Working Paper
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Abstract
We introduce a number of tools for finding and studying \emph{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 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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published
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1509.00632
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Cornell University
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v2
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09561 - Sisto, Alessandro (ehemalig) / Sisto, Alessandro (former)
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