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The Equivariant Coarse Novikov Conjecture and Coarse Embedding

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Abstract

The equivariant coarse Novikov conjecture provides an algorithm for determining nonvanishing of equivariant higher index of elliptic differential operators on noncompact manifolds. In this article, we prove the equivariant coarse Novikov conjecture under certain coarse embeddability conditions. More precisely, if a discrete group \(\Gamma \) acts on a bounded geometric space X properly, isometrically, and with bounded distortion, \(X/\Gamma \) and \(\Gamma \) admit coarse embeddings into Hilbert space, then the \(\Gamma \)-equivariant coarse Novikov conjecture holds for X. Here bounded distortion means that for any \(\gamma \in \Gamma \), \(\sup _{x\in Y} d(\gamma x,x)<\infty \), where Y is a fundamental domain of the \(\Gamma \)-action on X.

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Acknowledgments

The authors wish to thank the referees for many valuable and constructive suggestions. They also would like to thank Jintao Deng for many useful discussions, and for carefully reading the articles and making many helpful comments.

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Authors and Affiliations

  1. College of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, People’s Republic of China

    Benyin Fu

  2. College of Mathematics and Statistics, Chongqing University (at Huxi Campus), Chongqing, 401331, People’s Republic of China

    Xianjin Wang

  3. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Guoliang Yu

Authors
  1. Benyin Fu
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  2. Xianjin Wang
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  3. Guoliang Yu
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Correspondence to Xianjin Wang.

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Communicated by Y. Kawahigashi.

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Benyin Fu is supported by NSFC (Nos. 11871342, 11771143). Xianjin Wang is supported by NSFC (No. 11771061). Guoliang Yu is supported by NSF (Nos. 1564398, 1700021) and NSFC (No. 11420101001).

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Fu, B., Wang, X. & Yu, G. The Equivariant Coarse Novikov Conjecture and Coarse Embedding. Commun. Math. Phys. 380, 245–272 (2020). https://doi.org/10.1007/s00220-020-03754-9

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  • DOI: https://doi.org/10.1007/s00220-020-03754-9

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