Uniformizing dessins and Bely\i maps via circle packing
About this Title
Philip L. Bowers and Kenneth Stephenson
Publication: Memoirs of the American Mathematical Society
Publication Year 2004: Volume 170, Number 805
ISBNs: 978-0-8218-3523-4 (print); 978-1-4704-0406-2 (online)
MathSciNet review: 2053391
MSC (2000): Primary 30F10; Secondary 14H55, 30F50, 52C26
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 Belyibreve 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.
Graduate students and research mathematicians interested in geometry and topology.
Table of Contents
- 1. Introduction
- 2. Dessins d'Enfants
- 3. Discrete dessins via circle packing
- 4. Uniformizing dessins
- 5. A menagerie of dessins d'Enfants
- 6. Computational issues
- 7. Additional constructions
- 8. Non-equilateral triangulations
- 9. The discrete option
- 10. Appendix: Implementation