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Uniformizing Dessins and BelyĭMaps via Circle Packing
Philip L. Bowers, Florida State University, Tallahassee, FL, and Kenneth Stephenson, University of Tennessee, Knoxville, TN
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Memoirs of the American Mathematical Society
2004; 97 pp; softcover
Volume: 170
ISBN-10: 0-8218-3523-8
ISBN-13: 978-0-8218-3523-4
List Price: US$61
Individual Members: US$36.60
Institutional Members: US$48.80
Order Code: MEMO/170/805
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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 Belyĭ 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.

Readership

Graduate students and research mathematicians interested in geometry and topology.

Table of Contents

  • Introduction
  • Dessins d'enfants
  • Discrete Dessins via circle packing
  • Uniformizing Dessins
  • A menagerie of Dessins d'enfants
  • Computational issues
  • Additional constructions
  • Non-equilateral triangulations
  • The discrete option
  • Appendix: Implementation
  • Bibliography
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