Cyclic n-gonal Riemann surfaces, S, are of great interest since they are algebraic curves defined by y^n = f(x) for some polynomial f(x). For n=2 the surfaces are hyperelliptic surfaces, which are very well studied. The cases for small  n or n a prime are also well studied. The cyclic n-gonal  surface S has a cyclic group C  of automorphisms generalizing the notion of hyperelliptic involution. Under reasonable hypotheses the group C is a normal subgroup of the full group of automorphisms of S when the genus is large. If C is normal then the determination of the full automorphism then reverts to a careful analysis of the finite groups of automorphisms of the sphere.

In this talk we are going to extend the problem of determining the full automorphism group A of a cyclic n-gonal surface S under assumption of weak normality, i.e., that for any non-trivial H  < C, the normalizers of H and C in A are equal. This assumption automatically holds when n is a prime. Again in this case the group is normal when the genus is large enough with respect to  n. We will focus on the exceptional, low genus cases.