CAT(0) spaces quasi-isometric to Euclidean spaces

Abstract

We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space R^n then it is homeomorphic to R^n. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to R^n, which are not homeomorphic to it. We prove that our example is sharp in a suitable sense. Finally, we provide an example of a sequence of proper, geodesically complete, CAT(0) spaces that are not homology manifolds and that converge in the Gromov-Hausdorff sense to a topological manifold: this shows that the set of topological manifolds is not open in the class of proper, geodesically complete, CAT(0) spaces.