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arXiv · 1412.5828

On coarse geometric aspects of the Hilbert geometry

Ryosuke Mineyama·Shin-ichi Oguni
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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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Ryosuke Mineyama, Shin-ichi Oguni. 2014-12-18. On coarse geometric aspects of the Hilbert geometry. https://arxiv.org/abs/1412.5828

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