Abstract

Median spaces generalise R-trees and CAT cube complexes, and arise in nature as asymptotic cones of many different groups. One way to build median spaces from simple ones is to start with a (possibly infinite) product of trees, and delete “quarterspaces”. I will explain what this means, and why this shows that any CAT cube complex arises from the version of this construction where all of the trees are [0,1]. By replacing [0,1] with arbitrary R-trees, and allowing slightly more complicated rules specifying which quarterspaces one can delete, one obtains a class of median spaces called “R-cubings”, which generalise cube complexes and R-trees but have more structure than general median spaces. The utility of R-cubings is the following theorem: asymptotic cones of a hierarchically hyperbolic space (e.g. mapping class groups, right-angled Artin/Coxeter groups, Teichmuller space) are bilipschitz equivalent to R-cubings. I will discuss an application of this to the question of when asymptotic cones of a group are independent of the parameters used to define them and, time permitting, mention some work in progress about actions on R-cubings. This is all joint work with Montserrat Casals-Ruiz and Ilya Kazachkov.