Quasi-isometries between graphs and trees

Abstract

Criteria for quasi-isometry between trees and general graphs as well as for quasi-isometries between metrically almost transitive graphs and trees are found. Thereby we use different concepts of thickness for graphs, ends and end spaces. A metrically almost transitive graph is quasi-isometric to a tree if and only if it has only thin metric ends (in the sense of Definition 3.6). If a graph is quasi-isometric to a tree then there is a one-to-one correspondence between the metric ends and those d-fibers which contain a quasi-geodesic. The graphs considered in this paper are not necessarily locally finite. © 2007 Elsevier Inc. All rights reserved.

DOI: 10.1016/j.jctb.2007.06.008

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@article{Krn2008QuasiisometriesBG, title={Quasi-isometries between graphs and trees}, author={Bernhard Kr{\"{o}n and R{\"{o}gnvaldur G. M{\"{o}ller}, journal={J. Comb. Theory, Ser. B}, year={2008}, volume={98}, pages={994-1013} }