  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.
