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On coarse geometric aspects of Hilbert geometry

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Abstract

We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum–Connes conjecture holds for any two-dimensional Hilbert geometry via Yu’s theorem.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Ryosuke Mineyama

  2. Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790-8577, Japan

    Shin-ichi Oguni

Authors
  1. Ryosuke Mineyama
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  2. Shin-ichi Oguni
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Correspondence to Ryosuke Mineyama.

Additional information

Communicated by A. Constantin.

R. Mineyama: partly supported by Grant-in-Aid for JSPS Fellows No. 13J01771. S. Oguni: partly supported by JSPS Grant-in-Aid for Young Scientists (B) Nos. 24740045, 16K17595.

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Mineyama, R., Oguni, Si. On coarse geometric aspects of Hilbert geometry. Monatsh Math 187, 665–680 (2018). https://doi.org/10.1007/s00605-018-1171-1

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  • DOI: https://doi.org/10.1007/s00605-018-1171-1

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