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Riemann , Hurwitz , and Branched Covering Spaces

We will consider spaces with nice connectedness properties, and the groups that act on them in such a way that the topology is preserved; we consider looking at the symmetry groups of a surface of genus g. Restricting our view to finite groups, we will develop the concept of covering spaces and illustrate its usefulness to the study of these group actions by generalizing this development to the theory of branched coverings. This theory lets us develop the famous Riemann-Hurwitz Relation, which will in turn allow us to develop Hurwitz’s Inequality, an upper bound on the order of a symmetry group of a given surface. We then follow Kulkarni and use the Riemann-Hurwitz Relation to construct a congruence relating the genus g of a surface to the cyclic deficiencies of the symmetry groups that can act on it. These developments will then be applied to study a special case of branched coverings, those in which there is only one branch point, yielding a lower bound on the genus of both surfaces involved. Acknowledgments It was Dr. Robert Little whom gave me a paper that eventually lit a fire under a broiler in my mind, for that I thank him. I also truly appreciate the time he took to explain to me, in pictures and in words, those concepts that eluded me at the time. All his help is greatly appreciated. Thanks goes out to Dr. Mike Hilden for his notes and talks about branched coverings, his examples gave the conceptual clarity that I needed. For all the other Professors and Grad-Students who were there along the way, whether for a question about a complex mathematical problem or for a casual conversation during a run-in, I must thank them for the time that they gave me as much of my motivation and inspiration was drawn from these interactions. Most importantly, I give great thanks, and my heart, out to my wife. This accomplishment only became a reality because of her continual love and support. Aloha, mahalo nui loa.