Here is a working draft of the paper Real cubings and asymptotic cones of hierarchically hyperbolic groups, which is joint work with Montserrat Casals-Ruiz and Ilya Kazachkov.

This is the 29 March 2022 version. It contains Parts 1-3 of the paper.

Parts 1-3 develop the theory of real cubings, give a self-contained account of hierarchical hyperbolicity, show that asymptotic cones of hierarchically hyperbolic spaces are real cubings, and use this to describe the local structure of the asymptotic cones.

In Parts 4-7, this is applied to show uniqueness of certain cones; these parts are currently being tidied up, and will be posted here in the near future. The paper is long, so we have posted Parts 1-3 in the meantime, since they are self-contained and we are extremely interested in receiving comments on them.

The introduction will be expanded but gives some hints about Parts 4-7 as well as a more detailed discussion of Parts 1-3.