If a group G acts isometrically on a metric space X and Y is any metric space that is quasi-isometric to X, then G quasi-acts on Y. A fundamental problem in geometric group theory is to straighten (or quasi-conjugate) a quasi-action to an isometric action on a nice space. We will introduce and investigate discretisable spaces, those for which every cobounded quasi-action can be quasi-conjugated to an isometric action of a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups have this property, but it holds much more generally. For instance, we show that every hyperbolic group is either commensurable to a cocompact lattice in rank one Lie group, or it is discretisable.
We give several applications and indicate possible future directions of this ongoing work, particularly in showing that normal and almost normal subgroups are often preserved by quasi-isometries. For instance, we show that any finitely generated group quasi-isometric to a Z-by-hyperbolic group is Z-by-hyperbolic. We also show that within the class of residually finite groups, the class of central extensions of finitely generated abelian groups by hyperbolic groups is closed under quasi-isometries.[-]

Since the early 70s, Mackey and Green functors have been successfully used to model the induction and restriction maps which are ubiquitous in the representation theory of finite groups. In concrete examples, the latter maps are typically distilled, in some way, from induction and restriction functors between additive (abelian, triangulated...) categories. In order to better capture this richer layer of equivariant information with a (light!) set of axioms, we are naturally led to the notions of Mackey and Green 2-functors. Many such structures have been in use for a long time in algebra, geometry and topology. We survey examples and applications of this young—yet arguably overdue—theory. This is partially joint work with Paul Balmer and Jun Maillard.[-]