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Abstract
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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
is a closed irreducible
-manifold then
is an HHS if and
only if it is neither
nor
.
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.
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Keywords
geometric group theory, hierarchically hyperbolic, mapping
class group
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Mathematical Subject Classification 2010
Primary: 20F36, 20F65, 20F67
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Milestones
Received: 25 January 2018
Revised: 15 November 2018
Accepted: 27 November 2018
Published: 21 May 2019
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