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Some remarks on the simplicial volume of nonpositively curved manifolds

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Abstract

We show that any closed manifold with a metric of nonpositive curvature that admits either a single point rank condition or a single point curvature condition has positive simplicial volume. We use this to provide a differential geometric proof of a conjecture of Gromov in dimension three.

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Acknowledgements

The authors would like to thank Werner Ballmann and Jean-François Lafont for helpful discussions and the anonymous referees for both a careful scrutiny and suggesting several helpful clarifications.

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Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Chris Connell

  2. Max Planck Institute for Mathematics, 53111, Bonn, Germany

    Shi Wang

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  1. Chris Connell
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  2. Shi Wang
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Correspondence to Chris Connell.

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Communicated by Thomas Schick.

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Connell, C., Wang, S. Some remarks on the simplicial volume of nonpositively curved manifolds. Math. Ann. 377, 969–987 (2020). https://doi.org/10.1007/s00208-020-01987-6

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  • DOI: https://doi.org/10.1007/s00208-020-01987-6

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