Relating asymptotic dimension to Ponomarev's cofinal dimension via coarse proximities

TL;DR

This paper establishes bounds on the asymptotic dimension of unbounded proper metric spaces using a new coarse cofinal dimension concept, connecting it to topological and Higson corona dimensions through coarse proximity space constructions.

Contribution

It introduces the coarse cofinal dimension and inverse limits of coarse proximity spaces, linking asymptotic dimension with topological cofinal dimensions in a novel way.

Findings

Asymptotic dimension is bounded above by coarse cofinal dimension.

Asymptotic dimension is bounded below by cofinal dimension of Higson corona.

Introduces inverse limit construction in coarse proximity spaces.

Abstract

In this paper we show that the asymptotic dimension of an unbounded proper metric space is bounded above by a coarse analog of Ponomarev's cofinal dimension of topological spaces, which we call the coarse cofinal dimension. We also show that asymptotic dimension is bounded below by the cofinal dimension of the Higson corona by existing results of Miyata, Austin, and Virk. We do this by introducing several constructions in the theory of coarse proximity spaces. In particular we introduce the inverse limit of coarse proximity spaces. We end with some open problems.