Abstract and Applied Analysis

Hyperbolic Tessellation and Colorings of Trees

Dong Han Kim and Seonhee Lim

Full-text: Open access


We study colorings of a tree induced from isometries of the hyperbolic plane given an ideal tessellation. We show that, for a given tessellation of the hyperbolic plane by ideal polygons, a coloring can be associated with any element of Isom( 2 ), and the element is a commensurator of Γ if and only if its associated coloring is periodic, generalizing a result of Hedlund and Morse.

Article information

Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 706496, 6 pages.

First available in Project Euclid: 26 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)


Kim, Dong Han; Lim, Seonhee. Hyperbolic Tessellation and Colorings of Trees. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 706496, 6 pages. doi:10.1155/2013/706496. https://projecteuclid.org/euclid.aaa/1393449622

Export citation


  • M. Morse and G. A. Hedlund, “Symbolic dynamics II. Sturmian trajectories,” American Journal of Mathematics, vol. 62, pp. 1–42, 1940.
  • G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, vol. 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Berlin, Germany, 1991.
  • A. Kubena and A. Thomas, “Density of commensurators for uniform lattices of right-angled buildings,” Journal of Group Theory, vol. 15, no. 5, pp. 565–611, 2012.
  • N. Avni, S. Lim, and E. Nevo, “On commensurator growth,” Israel Journal of Mathematics, vol. 188, pp. 259–279, 2012.
  • V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, vol. 139 of Pure and Applied Mathematics, Academic Press, Boston, Mass, USA, 1994, translated from the 1991 Russian original by Rachel Rowen.
  • H. Bass and R. Kulkarni, “Uniform tree lattices,” Journal of the American Mathematical Society, vol. 3, no. 4, pp. 843–902, 1990.
  • H. Bass and A. Lubotzky, Tree Lattices, vol. 176 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 2001.
  • J.-P. Serre, Trees, Springer, Berlin, Germany, 1980.
  • N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, vol. 1794 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2002.