Quasi-isometries between graphs and trees

  • Authors:

  • Bernhard Krön;Rögnvaldur G. Möller

  • Affiliations:

  • University of Vienna, Faculty of Mathematics, Nordbergstraße 15, 1090 Wien, Austria;Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2008

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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.