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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.

MSC:
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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