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.[-]