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Generalized Hecke groups and Hecke polygons. (English) Zbl 0926.30026
From the abstract: “We study certain Fuchsian groups \({\mathcal H} (p_1, \dots, p_n)\), called generalized Hecke groups. These groups are isomorphic to \(\prod^{*n}_{j=1} Z_{p_j}\). Let \(\Gamma\) be a subgroup of finite index in \({\mathcal H} (p_1, \dots, p_n)\). By Kurosh’s theorem, \(\Gamma\) is isomorphic to \(F_r* \prod^{*k}_{i=k} Z_{m_i}\), where \(F_r\) is a free group of rank \(r\), and each \(m_i\) divides some \(p_j\). The signature of \(\Gamma\) is \((g;m_1, \dots, m_k; t)\), where \(g\) and \(t\) are the genus and the number of cusps of \(H^2/\Gamma\), respectively. The purpose of this paper is to consider two problems. First, determine the necessary and sufficient conditions for the existence of a subgroup of finite index of a given type in \({\mathcal H}(p_1, \dots, p_n)\). We also extend this work to extended generalized Hecke groups \({\mathcal H}^* (p_1, \dots, p_n)\) which are isomorphic to \(D_{p_1} *_{z_2} \cdots *_{z_2}D_{p_n}\) (amalgamated over \(Z_2\)’s generated by reflections), where each \(D_{p_j}\) is a dihedral group of order \(2p_j\). The second problem is the realizability problem for the existence of a subgroup with a given signature in \({\mathcal H}(p_1, \dots, p_n)\). This is a special case of the Hurwitz problem for the realizability of branched covers. Our approach is based on constructing special PoincarĂ© polygons which are the same as fundamental domains for \({\mathcal H}(p_1, \dots, p_n)\), \({\mathcal H}^* (p_1, \dots, p_n)\) and their subgroups”. The geometric constructions are quite explicit and several examples are worked through with fine illustrations.

30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
11F06 Structure of modular groups and generalizations; arithmetic groups
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