Bicombing the mapping class group and Teichm\"uller space via stable cubical intervals

TL;DR

This paper offers a new, simplified account of how mapping class groups and Teichmüller spaces admit bicombings, using hierarchical hyperbolic space structures and cubical complexes, with implications for geometric group theory.

Contribution

It introduces a novel, streamlined approach to bicombings in hierarchical hyperbolic spaces, connecting hulls to finite CAT(0) cube complexes with bounded dimension.

Findings

Hierarchical hulls are quasi-isometric to finite CAT(0) cube complexes.

Perturbations of point pairs cause bounded changes in the cubical structure.

The approach simplifies previous proofs and broadens understanding of geometric structures.

Abstract

In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichm\"uller 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.