In the context of CAT (0) cubical groups, we develop an analogue of the theory of curve
complexes and subsurface projections. The role of the subsurfaces is played by a collection …

As demonstrated by Croke and Kleiner, the visual boundary of a CAT (0) group is not well-
defined since quasi-isometric CAT (0) spaces can have non-homeomorphic boundaries. We …

We introduce two new tools for studying CAT (0) spaces: curtains, versions of cubical
hyperplanes; and the curtain model, a counterpart of the curve graph. These tools shed new …

This paper is a survey dedicated to the following question: given a group acting on a CAT (0)
cube complex, how to exploit this action to determine whether or not the group is …

Hierarchically hyperbolic spaces provide a common framework for studying mapping class
groups of finite-type surfaces, Teichmüller space, right-angled Artin groups, and many other …

We develop an analogy between right-angled Artin groups and mapping class groups
through the geometry of their actions on the extension graph and the curve graph …

Let χ be a proper CAT (0) cube complex admitting a proper cocompact action by a group G.
We give three conditions on the action, any one of which ensures that χ has a factor system …

We prove that all hierarchically hyperbolic groups have finite asymptotic dimension. One
application of this result is to obtain the sharpest known bound on the asymptotic dimension …

We define a bounded cohomology class, called the median class, in the second bounded
cohomology, with appropriate coefficients, of the automorphism group of a finite-dimensional …

The main technical result of this paper is to characterize the contracting isometries of a CAT
(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce …