Median metric spaces are a common generalisation of real trees and CAT(0) cube complexes. A real cubing is a median space arising as a subalgerba of a (generally infinite) product of real trees determined by a (generally infinite) system of consistency conditions on pairs of coordinates. When the real trees are unit intervals, one recovers CAT(0) cube complexes as examples of real cubings. The definition can also be seen as a fine-geometric analogue of the notion of a hierarchically hyperbolic space, with the real trees playing the role of "curve graphs" and the consistency conditions playing the analogous role to the Behrstock inequality.
Our first result says that any asymptotic cone of a hierarchically hyperbolic space is bilipschitz equivalent to a real cubing; this generalises/strengthens a result of Behrstock-Drutu-Sapir on mapping class groups. I will discuss the idea of this result, which relies on the theory of measured wallspaces. I will then discuss our main application, to uniqueness (up to bilipschitz equivalence) of asymptotic cones of various hierarchically hyperbolic groups, like mapping class groups and RAAGs.
This talk is on joint work in progress with Montserrat Casals-Ruiz and Ilya Kazachkov (working draft: https://www.wescac.net/cones.html).
