Morse coding for a Fuchsian group of finite covolume

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October 2009, 3(4): 637-646. doi: 10.3934/jmd.2009.3.637

Morse coding for a Fuchsian group of finite covolume

1.	Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received November 2009 Revised December 2009 Published January 2010

Abstract
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We consider a Fuchsian group Г and the factor surface H/Г, which has constant curvature $-1$ and maybe a few singularities. If we lift the surface continuously to $\H$ (except for a subset of a lower dimension), we obtain a fundamental domain $\D$ of Г. This can be done in different ways; ours is to restrict the choice to so-called Dirichlet domains, which always are convex polygonal subsets of $\H$. Given a generic geodesic on $\H$, one can produce a so-called geometric Morse code (or the cutting sequence) of the geodesic with respect to $\D$. We prove that the set of Morse codes of all generic geodesics on $\H$ with respect to $\D$ forms a topological Markov chain if and only if $\D$ is an ideal polygon.

Keywords: Fuchsian group, Morse coding of geodesics, Dirichlet domain., topological Markov chain, cutting sequence.

Mathematics Subject Classification: Primary: 37D40, 37B10; Secondary: 20H1.

Citation: Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637

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