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Mathematische Annalen

, Volume 373, Issue 1–2, pp 191–235 | Cite as

Ends of finite volume, nonpositively curved manifolds

  • Grigori AvramidiEmail author
Article

Abstract

We study complete, finite volume n-manifolds M of bounded nonpositive sectional curvature. A classical theorem of M. Gromov says that if such M has negative curvature then it is homeomorphic to the interior of a compact manifold-with-boundary, and we denote this boundary \(\partial M\). If \(n\ge 3\), we prove that the universal cover of the boundary \(\widetilde{\partial M}\) and also the \(\pi _1M\)-cover of the boundary \(\partial {{\widetilde{M}}}\) have vanishing \((n-2)\)-dimensional homology. For \(n=4\) the first of these recovers a result of Nguy\(\tilde{\hat{\mathrm {e}}}\)n Phan saying that each component of the boundary \(\partial M\) is aspherical. For any \(n\ge 3\), the second of these implies the vanishing of the first group cohomology group with group ring coefficients \(H^1(B\pi _1M;{\mathbb {Z}}\pi _1M)=0\). A consequence is that \(\pi _1M\) is freely indecomposable. These results extend to manifolds M of bounded nonpositive curvature if we assume that M is homeomorphic to the interior of a compact manifold with boundary. Our approach is a form of “homological collapse” for ends of finite volume manifolds of bounded nonpositive curvature. This paper is very much influenced by earlier, yet still unpublished work of Nguy\(\tilde{\hat{\mathrm {e}}}\)n Phan.

Notes

Acknowledgements

I would like to thank T\(\hat{\mathrm {a}}\)m Nguy\(\tilde{\hat{\mathrm {e}}}\)n Phan for explaining her proof of Theorem 2, discussing numerous aspects of the present paper and making crucial suggestions (e.g. the idea of how to organize patches into maximal clumps).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of MuensterMuensterGermany

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