CAT(0) spaces quasi-isometric to Euclidean spaces

2026/03/31 by Nicola Cavallucci, Andrea Sambusetti
Mathematics · #math.MG #math.DG #math.GT

paper · pdf

v2: Small modifications to the introduction

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We show that if a proper, geodesically complete, CAT(0) homology manifold is quasi-isometric to the Euclidean space Rⁿ then it is homeomorphic to Rⁿ. On the other hand, we show that there exist proper, geodesically complete, CAT(0) spaces quasi-isometric to Rⁿ, 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.

Citations

• CAT(0) 4-manifolds are Euclidean
• Continuity of critical exponent of quasiconvex-cocompact groups under Gromov-Hausdorff convergence
• Asymptotic topological regularity of CAT(0) spaces
• Packing and doubling in metric spaces with curvature bounded above
• Volume pinching theorems for CAT(1) spaces
• Topological regularity of spaces with an upper curvature bound
• Geodesically complete spaces with an upper curvature bound
• Notes on Pointed Gromov-Hausdorff Convergence
• The de Rham decomposition theorem for metric spaces
• Metric Spaces of Non-Positive Curvature
• The large scale geometry of nilpotent Lie groups
• An example of anomalous singular homology