Four-punctured spheres

Abstract

The four-punctured sphere can be viewed as a Riemann surface. It is then the quotient of the hyperbolic plane H by a Fuchsian group with signature [0; oo4]. In Chapter 1 we introduce Riemann surfaces and Fuchsian groups, in particular the subgroups of the modular group and Hecke groups. In Chapter 2 we introduce Teichmuller space via the Teichmuller space of the torus. We present the theory of Thompson [Th] and McNeilly [McNl] which gives a bijection between the Teichmuller space of groups of signature [0; oo4], denoted by T[0; oo4], and a domain V via a standard set of generators for such a group. We also describe the bijection between T[0; oo4] and T[l; oo] given by McNeilly In Chapter 3, the signatures of all Fuchsian groups Gi which contain a group G\ of signature [0; oo4] with finite index are found. For each such group inclusion G\ C G2 we find generators for the groups G\ and G2, and using the theory of Chapter 2 find the corresponding points in T[0; oo4]. The interesting case is when Gi is a triangle group in which case we say the corresponding G\ is exceptional. We also find the groups containing a subgroup of signature [1; 00] with finite index and the corresponding points in T[l; 00]. We introduce the Teichmuller modular group of a group of signature [0; oo4], and study its action on T[0; oo4]. We find that exceptional groups are congruence subgroups of the modular group and have rational points in V. We also give a short proof of a Theorem of Thompson [Th]. Finally, in Chapter 4 we introduce NEC groups and consider those NEC groups that have a group of signature [0; oo4] as their canonical Fuchsian group. Corresponding to these groups are some one-dimensional subspaces of T[0; oo4] which we calculate. We conclude with a brief discussion of further work related to onedimensional subspaces.

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