| We describe a conjectural characterization of all groups quasi-isometric to H x Rn, where H is any non-elementary hyperbolic group, and we provide an outline of the steps required to establish such a characterization. We carry out several steps of this plan. We consider those lines Lˆ in the asymptotic cone Coneo(H) which, in a precise sense, "arise from lines L in H''. We give a complete description of such lines, showing (in particular) that they are extremely rare in Coneo(H). Given a top-dimensional quasi-flat in H x R n, we show the induced bi-Lipschitz embedded flat in Coneo(H x Rn) must lie uniformly close to some Lˆ x Rn, where Lˆ is one of these rare lines. As a result, we conclude that quasi-actions on H x R n must project to quasi-actions on H and therefore to homeomorphic actions on ∂H. Finally, we show that such an action on ∂H is a convergence action which is uniform if it is discrete, and we discuss the work that remains to complete the conjectured characterization. |
