Uniformizing dessins and Belyĭ maps via circle packing
Philip L. Bowers, Kenneth Stephenson · Memoirs of the American Mathematical Society · 2004
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. The paper goes on to show that the discrete dessin structures converge to their classical counterparts under a hexagonal refinement scheme. In addition, since the discrete objects are computable, circle packing provides opportunities both for routine experimentation and for large scale explicit computation. A range of examples up to genus 4 is given in the paper, and an a...