Abstract

A fundamental problem in geometric group theory is the study of quasi-actions. We introduce and investigate discretisable spaces: spaces for which every cobounded quasi-action can be quasi-conjugated to an isometric action on a locally finite graph. Work of Mosher-Sageev-Whyte shows that free groups are discretisable spaces, but the property holds much more generally. For instance, every non-elementary hyperbolic group is either virtually isomorphic to a cocompact lattice in rank one Lie group, or it is discretisable.

Along the way, we introduce the concept of the topological completion of a quasi-action. This is a locally compact group, well-defined up to a compact normal subgroup, reflecting the geometry of the quasi-action. We give several applications of the tools we develop. For instance we show that any finitely generated group quasi-isometric to a ‬Z‭-by-hyperbolic group is also Z-by-hyperbolic, and prove quasi-isometric rigidity for a large class of right-angled Artin groups.