## Classifying Pairs of Fuchsian Groups
of Finite Type

### - preliminary report-

at the

### ** University of Arizona, Tucson, AZ (2007 Spring Western Section Meeting)
Meeting # 1027**

### Abstract

A pair of finite type Fuchsian groups is often part of the answer to some
questions in the theory of Riemann surfaces. First, if a Riemann surface has
a known group of automorphisms is the group the full automorphism group of
the surface? Second, the moduli space of surfaces of fixed genus may be stratified
into a disjoint union of smooth subvarieties such that the automorphism
group of each surface is topologically constant along the strata. When does
one stratum lie in the closure of another? Third, the hyperbolic plane may be
tiled by reflections in the sides of certain polygons. When does the tiling admit
a refinement by a tiling generated by a polygon of smaller area? In each of
these questions the answer is determined by a pair of Fuchsian groups $\Gamma \subset \Delta$ that
satisfies certain algebraic properties. This report gives some preliminary classification results on
pairs when the difference of the Teichmueller dimensions
(codimension) of the groups is small. The codimension zero case was determined
some years ago in the context of classifying finitely maximal Fuchsian groups.
### Lecture and other materials

### Links

Email: allen.broughton@rose-hulman.edu
Webpage: http://www.rose-hulman.edu/~brought/
This page last updated on 22 Apr 07.