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Stable cubulations, bicombings, and barycenters

Matthew G Durham, Yair N Minsky and Alessandro Sisto

Geometry & Topology 27 (2023) 2383–2478
Abstract

We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichmüller spaces are stably approximated by CAT(0) cube complexes, strengthening a result of Behrstock, Hagen and Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichmüller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of “colorable” hierarchically hyperbolic spaces and groups.

Keywords
semihyperbolic, bicombing, mapping class group, hierarchically hyperbolic, cube complexes, Teichmüller space
Mathematical Subject Classification
Primary: 20F65, 57K20
References
Publication
Received: 3 June 2021
Revised: 11 November 2021
Accepted: 12 December 2021
Published: 25 August 2023
Proposed: Mladen Bestvina
Seconded: David Fisher, Benson Farb
Authors
Matthew G Durham
Department of Mathematics
University of California Riverside
Riverside, CA
United States
Yair N Minsky
Department of Mathematics
Yale University
New Haven, CT
United States
Alessandro Sisto
Department of Mathematics
Heriot-Watt University
Edinburgh
United Kingdom

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