Limit sets of Kleinian groups: properties, parameters, andpictures Open Access

Abstract

This thesis explores fractal images that are the limit sets of Kleinian groups. Properties and classification of Möbius transformations lead to groups of such transformations, and to the

classification of some important group types. Möbius transformations are represented by matrices in SL(2, C). The purpose of the early material is to foster an intuitive grasp of what

happens in simple groups, so that more complex groups may be correctly pictured. The accumulation points of group's actions on points in the upper half space are on the plane and are

called the limit set. The limit set of a Kleinian group can have 0, 1, 2, or uncountably infinitely many points. A discrete group will have a limit set that is a proper subset of the plane. The limit set is the smallest nonempty, closed subset of the plane that is invariant under the action of the group. Besides properties of the limit set, the thesis explores and utilizes a depth-first search to plot pictures of connected limit sets. The family used for the algorithm has connected limit sets that are origin-symmetric and pass through 1 and -1. Each group in the family has a parabolic commutator. The pictures are used to explore different features of the groups in the family--ones generated by parabolic, hyperbolic, and loxodromic transformations. The images, generated by Mathematica, by self-symmetry and spirals, serve to remind readers of facts about complex arithmetic and Möbius transformations. The final chapter also explores some limitations of the algorithm.

Table of Contents

Introduction 1

Chapter 1: Möbius transformations 4

Chapter 2: Groups of transformations 12

Chapter 3: Limit sets 23

Chapter 4: Graphing the limit set 31

References: 43

Appendix: the code 44

About this Honors Thesis

Rights statement

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School	Emory College
Department	Mathematics
Degree	BS
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	conjugation Limit sets torsion-free algorithm Kleinian groups Parabolic Moebius transformations commutator Mobius transformations
Committee Chair / Thesis Advisor	Borthwick, David, Emory University
Committee Members	Berland, Keith, Emory University Hamilton, Emily, Emory University

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