Normal subgroups of Hecke groups

Abstract

The Hecke group H(λq) is the discrete subgroup of PSL(2,R) generated by R(z) = -1/z and T(z) = z + λq for λq = 2cosπ/q. In this thesis we investigate normal subgroups of these Hecke groups.

The most important Hecke group is the modular group obtained for q = 3. There are many results in the literature concerning normal subgroups of the modular group. We are interested in generalizing these to other Hecke groups H(λq).

Jones and Singerman, [Jo-Si,1], determined a 1:1 correspondence between normal subgroups of certain triangle groups, including Hecke groups, and regular maps. We study normal subgroups of H(λq) by means of this correspondence and obtain results about normal subgroups of H(λq) using the known regular maps. This is especially useful when g = 0 or 1, as all regular maps with these genera are classified (see [Co-Mo,1] or [Jo-Si,1]).

We obtain fairly complete information about the normal subgroups of H(λ4), H(λ5) and H(λ6) of low index and obtain some other results for other values of q. In particular we investigate principal congruence subgroups of H(λq) for prime powers q. (DX183755)

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