Classifying Pairs of Fuchsian Groups of Finite Type

- preliminary report-

S. Allen Broughton
Rose-Hulman Institute of Technology
April 21, 2007

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

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