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Uniformizing dessins and Belyĭ maps via circle packing. (English) Zbl 1080.52511

Mem. Am. Math. Soc. 805, 97 p. (2004).
Summary: We study the structures on compact surfaces determined combinatorially in Grothendieck’s theory of dessins d’enfants; namely, affine, reflective and conformal structures. A parallel discrete theory is introduced based on circle packings and is shown to be geometrically faithful, even at its coarsest stages, to the classical theory. The resulting discrete structures converge to their classical counterparts under a hexagonal refinement scheme. In particular, circle packing offers a general approach for uniformizing dessin surfaces and approximating their associated Belyĭ meromorphic functions.

MSC:

52C26 Circle packings and discrete conformal geometry
30F10 Compact Riemann surfaces and uniformization
30C62 Quasiconformal mappings in the complex plane
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