Stable cubulations, bicombings, and barycenters

Matthew G. Durham, Yair N. Minsky, Alessandro Sisto

Published 2020-09-28Version 1

We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichm\"uller 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 Teichm\"uller 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.