**Memoirs of the American Mathematical Society**

2004;
97 pp;
Softcover

MSC: Primary 52; 30;

Print ISBN: 978-0-8218-3523-4

Product Code: MEMO/170/805

List Price: $65.00

Individual Member Price: $39.00

**Electronic ISBN: 978-1-4704-0406-2
Product Code: MEMO/170/805.E**

List Price: $65.00

Individual Member Price: $39.00

# Uniformizing Dessins and BelyĭMaps via Circle Packing

Share this page
*Philip L. Bowers; Kenneth Stephenson*

Grothendieck's theory of Dessins d'Enfants involves
combinatorially determined affine, reflective, and conformal structures
on compact surfaces. In this paper the authors establish the first
general method for uniformizing these dessin surfaces and for
approximating their associated Bely&ibreve; meromorphic functions.

The paper begins by developing a discrete theory
of dessins based on circle packing. This theory is surprisingly
faithful, even at its coarsest stages, to the geometry of the
classical theory, and it displays some new sources of richness; in
particular, algrebraic number fields enter the theory in a new
way. Furthermore, the discrete dessin structures converge to their
classical counterparts under a hexagonal refinement scheme. Since the
discrete objects are computable, circle packing provides opportunities
both for routine experimentation and for large scale explicit
computation, as illustrated by a variety of dessin examples up to
genus 4 which are computed and displayed.

The paper goes on to discuss uses of discrete
conformal geometry with triangulations arising in other situations,
such as conformal tilings and discrete meromorphic functions. It
concludes by addressing technical and implementation issues and open
mathematical questions that they raise.

#### Table of Contents

# Table of Contents

## Uniformizing Dessins and BelyiMaps via Circle Packing

- Contents v6 free
- List of Tables ix10 free
- List of Figures xi12 free
- Chapter1.Introduction 114 free
- Chapter 2. Dessins d'Enfants 720 free
- Chapter 3. Discrete Dessins via Circle Packing 1326
- Chapter 4. Uniformizing Dessins 2336
- Chapter 5. A Menagerie of Dessins d'Enfants 3750
- Chapter 6. Computational Issues 5568
- Chapter 7. Additional Constructions 6174
- Chapter 8. Non-equilateral Triangulations 7790
- Chapter 9. The Discrete Option 8396
- Chapter 10. Appendix: Implementation 89102
- Bibliography 95108

#### Readership

Graduate students and research mathematicians interested in geometry and topology.