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Coarse Quasi-Isometries

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Generic Coarse Geometry of Leaves

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2223))

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Abstract

This chapter is devoted to the study of coarse quasi-isometries. Of particular interest is the coarse version of the Arzelà-Ascoli theorem.

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Notes

  1. 1.

    This is slightly different from the definition of this concept given in [85].

  2. 2.

    Recall that M is called proper if its closed balls are compact.

  3. 3.

    This is also slightly different from the definition of this concept given in [20].

  4. 4.

    The definition of K-net is slightly different from the definition used in [9]. Our arguments become simpler in this way.

  5. 5.

    This terminology is used in [66]. Other terms used to indicate the same property are coarsely equivalent [85], parallel [56], bornotopic [84], and uniformly close [20].

  6. 6.

    Continuity is not assumed here.

  7. 7.

    This name is taken from [85]. Other terms used to denote the same property are uniformly bornologous [84] and coarsely Lipschitz [20].

  8. 8.

    This term is used in [85].

  9. 9.

    This term is used in [85]. Another term used to denote the same property is effectively proper [20].

  10. 10.

    This is a particular case of coarse maps between general coarse spaces [66, 85].

  11. 11.

    The term uniform closeness is used in [20] when two metric spaces are roughly equivalent.

  12. 12.

    This notion of metric coarse space is equivalent to the concept of coarse space induced by a metric [85, 86].

  13. 13.

    This is a subcategory of the coarse category [66, 85].

  14. 14.

    This concept is generalized to arbitrary coarse spaces as maps that define a coarse equivalence to their image [86, Section 11.1].

References

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Author information

Authors and Affiliations

  1. Department and Institute of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain

    Jesús A. Álvarez López

  2. Department of Mathematics, California State University, Northridge, California, USA

    Alberto Candel

Authors
  1. Jesús A. Álvarez López
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  2. Alberto Candel
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Álvarez López, J.A., Candel, A. (2018). Coarse Quasi-Isometries. In: Generic Coarse Geometry of Leaves. Lecture Notes in Mathematics, vol 2223. Springer, Cham. https://doi.org/10.1007/978-3-319-94132-5_2

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