Automorphism groups on compact Riemann surfaces

Abstract:

For $g \geqq 2$, let $N(g)$ be the order of the largest automorphism group on a Riemann surface of genus $g$. In this paper, lower bounds for $N(g)$ for various sequences of $g$’s are obtained. Sequences of appropriate groups are constructed. Each of these groups is then realized as a group of cover transformations of a surface covering the Riemann sphere. The genus of the resulting surface is then found by using the Riemann-Hurwitz formula and the automorphism group of the surface contains the given group. Each lower bound which is found is also shown to be sharp. That is, there are infinitely many $g$’s in the sequence to which the bound applies for which $N(g)$ does not exceed the bound.