Abstract

The theory of real trees and groups acting on them has had a deep impact on group theory: by providing tools to attack new problems, simplifying proofs, and establishing new connections between group theory and geometry, topology, dynamical systems, and model theory.

In this talk, we will introduce a new class of metric spaces, called real cubings, which we regard as higher-dimensional real trees, and characterise them inside the class of median spaces. As hyperbolic groups are linked to real trees via their asymptotic cones, we will show that real cubings are related to hierarchically hyperbolic groups, a class of groups that contains right-angled Artin groups and the mapping class groups of closed surfaces.

We will then describe the structure of universal real cubings and prove that there exists a group that has a free and transitive action on a given universal real cubing.

This is joint work with Montse Casals-Ruiz and Mark Hagen.