Discretisable quasi-actions I: Topological completions and hyperbolicity

Abstract

We define and develop the notion of a discretisable quasi-action. It is shown that a cobounded quasi-action on a proper non-elementary hyperbolic space $X$ not fixing a point of $\partial X$ is quasi-conjugate to an isometric action on either a rank one symmetric space or a locally finite graph. Topological completions of quasi-actions are also introduced. Discretisable quasi-actions are used to give several instances where commensurated subgroups are preserved by quasi-isometries. For example, the class of $\mathbb{Z}$-by-hyperbolic groups is shown to be quasi-isometrically rigid. We characterise the class of finitely generated groups quasi-isometric to either $\mathbb{Z}^n\times \Gamma_1$ or $\Gamma_1\times \Gamma_2$, where $\Gamma_1$ and $\Gamma_2$ are non-elementary hyperbolic groups.