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arXiv · 2603.23768

CAT(0) spaces quasi-isometric to Euclidean spaces

Nicola Cavallucci·Andrea Sambusetti
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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.

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Nicola Cavallucci, Andrea Sambusetti. 2026-03-24. CAT(0) spaces quasi-isometric to Euclidean spaces. https://arxiv.org/abs/2603.23768

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