On coarse geometric aspects of the Hilbert geometry

TL;DR

This paper explores the coarse geometric properties of Hilbert geometries, establishing conditions for their boundaries to be coronas, and demonstrating that certain conjectures hold for these geometries, especially in two dimensions.

Contribution

It provides necessary and sufficient conditions for the boundary to be a corona and proves the coarse Novikov and Baum-Connes conjectures for two-dimensional Hilbert geometries.

Findings

Boundary of Hilbert geometry is a corona under specific conditions.

Hilbert geometries are uniformly contractible with coarse bounded geometry.

Coarse Novikov and Baum-Connes conjectures hold for two-dimensional Hilbert geometries.

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.