# A combinatorial approach to coarse geometry - Mathematics > Metric Geometry

A combinatorial approach to coarse geometry - Mathematics > Metric Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: Using ideas from shape theory we embed the coarse category of metric spacesinto the category of direct sequences of simplicial complexes with bonding mapsbeing simplicial. Two direct sequences of simplicial complexes are equivalentif one of them can be transformed to the other by contiguous factorizations ofbonding maps and by taking infinite subsequences. That embedding can berealized by either Rips complexes or analogs of Roe-s anti-\v\{C}echapproximations of spaces.In that model the asymptotic dimension being at most n means that for each kthere is m > k such that the bonding map from K k to K m factors up tocontiguity through an n-dimensional complex. One can give a similarcharacterization of Property A of G.Yu. Using our approach we give a simpleproof of a characterization of geodesic spaces that are coarsely equivalent tosimplicial trees a result of Fujiwara and Whyte.

Author: ** M.Cencelj, J.Dydak, A.Vavpetiv{c}, v{Z}.Virk**

Source: https://arxiv.org/