Riemann Surfaces and 3-Regular Graphs

In this thesis we consider a way to construct a rich family of compact Riemann Surfaces in a combinatorial way. Given a 3-regualr graph with orientation, we construct a finite-area hyperbolic Riemann surface by gluing triangles according to the combinatorics of the graph. We then compactify this surface by adding finitely many points. We discuss this construction by considering a number of examples. In particular, we see that the surface depends in a strong way on the orientation. We then consider the effect the process of compactification has on the hyperbolic metric of the surface. To that end, we ask when we can change the metric in the horocycle neighbourhoods of the cusps to get a hyperbolic metric on the compactification. In general, the process of compactification can have drastic effects on the hyperbolic structure. For instance, if we compactify the 3-punctured sphere we lose its hyperbolic structure. We show that when the cusps have lengths > 2\pi, we can fill in the horocycle neighbourhoods and retain negative curvature. Furthermore, the last condition is sharp. We show by examples that there exist curves arbitrarily close to horocycles of length 2\pi, which cannot be so filled in. Such curves can even be taken to be convex.