A combinatorial approach to coarse geometry

TL;DR

This paper introduces a new combinatorial framework for coarse geometry by embedding metric spaces into sequences of simplicial complexes, providing simplified characterizations of properties like asymptotic dimension and Property A.

Contribution

It develops an embedding of the coarse category into sequences of simplicial complexes, enabling new characterizations of coarse properties and a simplified proof of a known theorem.

Findings

Embedding of coarse spaces into simplicial complexes via Rips and Roe's complexes

Characterization of asymptotic dimension through factorization of bonding maps

Simplified proof of geodesic spaces being coarsely equivalent to trees

Abstract

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. That embedding can be realized by either Rips complexes or analogs of Roe's anti-\v\{C}ech approximations of spaces. In that model the asymptotic dimension being at most n means that for each k there is m > k such that the bonding map from K_k to K_m factors (up to contiguity) through an n-dimensional complex. One can give a similar characterization of Property A of G.Yu. Using our approach we give a simple proof of a characterization of geodesic spaces that are coarsely equivalent to simplicial trees (a result of Fujiwara and Whyte).