Uniformizing Dessins and Belyi Maps via Circle Packing: July 2004, Volume 170, Number 805 (second of 4 numbers)

Bowers, Philip L.; and Kenneth Stephenson

ISBN 10: 0821835238 / ISBN 13: 9780821835234

Editore: American Mathematical Society, Providence, RI, 2004

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Two tiny holes in front cover front staple removal. Else covers clean. Two tiny indentations from same removed staple on first page. One line highlighted on copyright page. Else pages clean. Number 805 of the "Memoirs of the American Mathematical Society" series. ; MAT12D; 97 pages.

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Dati bibliografici

• Titolo: Uniformizing Dessins and Belyi Maps via Circle Packing: July 2004, Volume 170, Number 805 (second of 4 numbers)
• Editore: American Mathematical Society, Providence, RI
• Data di pubblicazione: 2004
• Lingua: Inglese
• ISBN 10: 0821835238
• ISBN 13: 9780821835234
• Rilegatura: Softcover
• Edizione: First Printing.
• Condizione: Very Good
• Condizione della sovraccoperta: No Dust Jacket

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Riassunto

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 Belyi 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.