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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board Member: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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1615-7168
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Volume 6, Issue 2 (Mar 2006)

Issues

Geodesics in non-positively curved plane tessellations

Oliver Baues
  • Math. Institut II, Universität Karlsruhe, 76128 Karlsruhe, Germany.
  • Email:
/ Norbert Peyerimhoff
  • Department of Math. Sciences, Durham University, Durham DH1 2LE, UK.
  • Email:
Published Online: 2006-05-17 | DOI: https://doi.org/10.1515/ADVGEOM.2006.014

Abstract

We introduce a natural combinatorial curvature function on the corners of plane tessellations and relate it to the global metric geometry of their corresponding edge and dual graphs. If the combinatorial curvature in the corners is non-positive then we prove that any geodesic path in such a graph may be extended to infinity. Moreover, if the combinatorial curvature is negative we show that every pair of geodesic segments with the same end points does not enclose any vertices. We apply these results to establish an estimate for the growth of distance balls, Gromov hyperbolicity, and four-colourability of certain classes of plane tessellations.

About the article


Received: 2004-04-27

Revised: 2005-04-27

Published Online: 2006-05-17

Published in Print: 2006-03-24


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/ADVGEOM.2006.014. Export Citation

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