S. Allen Broughton, Rose - Hulman Institute of
Technology (presenter)

Aaron Wootton, University of Portland (coauthor) for the first talk

February 26 - March 5, 2009

First talk 26 Feb 09.

**Title**: Full Automorphism Groups of Cyclic *n*-gonal Surfaces

(joint work with Aaron Wootton)

**Abstract:** 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.

- Slides of the talk (PDF file 570 K)
- paper (forthcoming)

Second talk 2 Mar 09.

**Title**: Classification of Pairs of Fuchsian Groups

**Abstract:** For 2-3 decades many paper's on automorphism of Riemann surfaces have referred to "Singerman's list" of
pairs of finitely maximal Fuchsian groups. The list has been especially useful in determining when a given
groups of automorphisms is the full automorphism. of a surface. Singerman's list may also be interpreted as the
the list of Fuchsian group pairs *H < G* where *H* and *G* have the same Teichmuller dimension.

In this talk we discuss the extension of the classification to pairs of Fuchsian group pairs *H < *G where the
Teichmuller codimension of *H* exceed that of *G* by a small integer *d(H,G)* called the Teichmuller codimension.
The main result is that for each fixed codimension there is a finite number of exceptional pairs and a finite number
of families. This neatly generalizes Singerman's result. Applications of the classification to extending
actions of groups and the analysis of the branch locus of moduli space will be discussed.

- Slides of the talk (PDF file 836 K)
- paper (forthcoming)

Email: allen.broughton@rose-hulman.edu Webpage: http://www.rose-hulman.edu/~brought/ This page last updated on 2 Mar 09.