Four-punctured spheres

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Description/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.