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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 Poincaré 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
##### Keywords:
generalizations of the modular group
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