Morse coding for a Fuchsian group of finite covolume (Q962235)

Let \(H\) denote the hyperbolic plane with Gaussian curvature \(-1\). Let \(\Gamma\) denote a discrete, torsion free subgroup of \(I(H)\), the isometries of \(H\), such that \(H/\Gamma\) has finite area. Let \(D\subset H\) denote a Dirichlet fundamental domain for the action of \(\Gamma\). A generic geodesic \(\gamma\) of \(H\) by definition meets none of the vertices of \(\{g(D): g\in\Gamma\}\). Such a geodesic \(\gamma\) generates a doubly infinite sequence \(\{h_k\}\subset \Gamma\) such that each \(h_k\) lies in \(\Sigma= \{g_1,\dots, g_N\}\), where \(\{g_1(D),\dots, g_N(D)\}\) are those \(\Gamma\)-copies of \(D\) that share an edge with \(D\). This encoding of the generic geodesics of \(H\) is due to M. Morse. Let \(\Lambda\) be the subset in \(\Sigma^{\mathbb Z}\) that arises from the encoded generic geodesics of \(H\). The set \(\Lambda\) is called a topological Markov chain if there exists a positive integer \(k\) and a subset \(\Lambda_k\) of \(\Sigma^{k+1}\) such that \(\Lambda= \{\lambda\in \Sigma^{\mathbb Z}: (\lambda_n,\dots, \lambda_{n+k})\in \Lambda_k\) for all \(n\in\mathbb Z\}\). In the case that \(\Gamma\) is the modular group \(E\), Artin developed another approach to encode the geodesics \(\gamma\) of \(H\) that uses the endpoints of \(\gamma\) to define real numbers that are then expanded into continued fractions. This method was later extended to arbitrary lattices \(\Gamma\) and modified so that the \(\Gamma\)-encoded geodesics of \(H\) produce topological Markov chains. The main result of the article is that the set \(\Lambda\) of \(\Sigma^{\mathbb Z}\) determined by the Morse encoding of generic geodesics in \(H\) is a topological Markov chain \(\Leftrightarrow\) the Dirichlet fundamental domain \(D\) has no finite vertices. In an earlier work, C. Series essentially proved one direction of the main result, namely, that if \(D\) has no finite vertices, then \(\Lambda\) is a topological Markov chain.