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.

Keywords

Hilbert geometry Corona Coarse Baum–Connes conjecture Coarse Novikov conjecture Asymptotic dimension 

Mathematics Subject Classification

51F99 

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Copyright information

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

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan
  2. 2.Department of Mathematics, Faculty of ScienceEhime UniversityMatsuyamaJapan

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