Hyperbolic Tessellation and Colorings of Trees

Abstract

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(${ℍ}^2$), and the element is a commensurator of $\mathrm{\Gamma }$ if and only if its associated coloring is periodic, generalizing a result of Hedlund and Morse.