Uniformizing dessins and Belyi maps via circle packing

Phil Bowers, Ken Stephenson

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 Belyi meromorphic functions. (64 pages with 3 tables and 28 figures)