Bicombing the mapping class group and Teichmüller space via stable cubical intervals

In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichmüller spaces admit bicombings. More generally, we explain how the hierarchical hull of a pair of points in any colorable hierarchically hyperbolic space is quasi-isometric to a finite CAT(0) cube complex of bounded dimension, with the added property that perturbing the pair of points results in a uniformly bounded change to the cubical structure. Our approach is simplified and new in many aspects.