Some calculations on the action of groups on surfaces

In this thesis we treat a number of topics related to generation of finite groups with motivation from their action on surfaces. The majority of our findings are presented in two chapters which can be read independently. The first deals with Beauville groups which are automorphism groups of the product of two Riemann surfaces with genus g > 1, subject to some further conditions. When these two surfaces are isomorphic and transposed by elements of G we say these groups are mixed, otherwise they are unmixed. We first examine the relationship between when an almost simple group and its socle are unmixed Beauville groups and then go on to determine explicit examples of several infinite families of mixed Beauville groups. In the second we determine the Mobius function of the small Ree groups 2G2(32m+1) = R(32m+1), where m >0, and use this to enumerate various ordered generating n-tuples of these groups. We then apply this to questions of the generation and asymptotic generation of the small Ree groups as well as interpretations in other categories, such as the number of regular coverings of a surface with a given fundamental group and whose covering group is isomorphic to R(32m+1)