CAT(0) spaces quasi-isometric to Euclidean spaces

TL;DR

This paper investigates the topological structure of CAT(0) spaces that are quasi-isometric to Euclidean spaces, establishing conditions under which they are homeomorphic to Euclidean spaces and providing counterexamples.

Contribution

It proves that certain CAT(0) homology manifolds quasi-isometric to Euclidean space are homeomorphic to it, and constructs examples showing the limits of this property.

Findings

Proper CAT(0) homology manifolds quasi-isometric to R^n are homeomorphic to R^n.

Existence of CAT(0) spaces quasi-isometric but not homeomorphic to R^n.

The set of topological manifolds is not open in the space of CAT(0) 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.