Generalized Hecke Groups and Hecke Polygons

In this paper, we study certain Fuchsian groups H (p1, . . . , pn) , called generalized Hecke groups. These groups are isomorphic to ∏∗ n j=1Zpj . Let Γ be a subgroup of finite index in H (p1, . . . , pn) . By Kurosh’s theorem, Γ is isomorphic to Fr ∗ ∏∗ k i=1Zmi , where Fr is a free group of rank r , and each mi divides some pj . Moreover, H/Γ is Riemann surface. The numbers m1, . . . , mk are branching numbers of the branch points on H/Γ. The signature of Γ is (g;m1 , . . . , mk; t) , where g and t are the genus and the number of cusps of H/Γ, respectively. A 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 H (p1, . . . , pn) . We also extend this work to extended generalized Hecke groups H (p1, . . . , pn) which are isomorphic to Dp1 ∗Z2 · · · ∗Z2 Dpn (amalgamated over Z2 ’s generated by reflections), where each Dpj is a dihedral group of order 2pj . The second problem is the realizability problem for the existence of a subgroup with a given signature in H (p1, . . . , pn) . This is a special case of the Hurwitz problem about the realizability of branched covers. Special cases of this work were also studied by Millington, Singerman, Hoare, Edmonds, Ewing and Kulkarni. Our approach is based on constructing special Poincaré polygons which are the same as fundamental domains for H (p1, . . . , pn) , H (p1, . . . , pn) and their subgroups.