| We first discuss the bijection of categories of compact Riemann surfaces with finitely generated field extensions K ⊃ C of transcendence degree one over C , and projective algebraic curves. Because of this bijection, we are able to use the techniques and methods of each category to explore the question of classifying monodromy groups. We then consider indecomposable degree d covers of Riemann surfaces. Next we discuss how the geometric questions of a monodromy group can be interpretted as an ordered triple G, H, E of group theoretic information, where G is a monodromy group, H is a maximal subgroup of G, and E is an ordered r-tuple with elements in G, that generates G. Previous work on classifying the monodromy groups is then discussed, particularly the Guralnick-Thompson Conjecture on the composition factors of a monodromy group G of genus g. After this we discuss the recent results of Guralnick and Shareshian on degree d covers with at least five branch points and alternating or symmetric group of degree n as monodromy group. We then explain their methods, and how to carry those methods further, in order to prove that if the cover has four branch points, then the genus grows rapidly with d unless either d = n or d = n 2 . If d = n 2 then the cycle shapes of elements of the generating set E are explicitly determined, confirming a conjecture of Guralncik and Shareshian. |
