Morse coding for a Fuchsian group of a finite covolume

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We consider the Morse coding of the geodesic flow on the hyperbolic plane H with respect to a Dirichlet fundamental domain D of a Fuchsian group Γ. The main theorem states that the codes of all the generic geodesics constitute a k-step topological Markov chain, if and only if the fundamental domain D is an ideal polygon (i.e. has all of its vertices on the absolute).