Stable cubulations, bicombings, and barycenters

Abstract

We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichmüller spaces are stably approximated by a CAT(0) cube complexes, strengthening a result of Behrstock-Hagen-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.

80 pages, 25 figures

References in corpus (4)

• Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity
• Stable Subgroups of the Genus Two Handlebody Group
• Hyperbolic HHS II: Graphs of hierarchically hyperbolic groups
• Hierarchical hyperbolicity of hyperbolic-2-decomposable groups

Cited by in corpus (4)

• Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity
• Extensions of Veech groups II: Hierarchical hyperbolicity and quasi-isometric rigidity
• The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover
• A Combinatorial Structure for Many Hierarchically Hyperbolic Spaces