Circle packings and combinatorial moduli
Several approaches have been developed to estimate the moduli of curves in topological spaces provided with very weak structures. The object of this text is to unify, as far as possible, the approaches of J.~W.~Cannon, W.~Floyd and W.~Parry and of M.~Bonk and B.~Kleiner to the uniformization problem of surfaces motivated by the conjecture of Cannon that a hyperbolic group whose boundary is homeomorphic to \(\mathbb S^2\) admits an Klein cocompact action. These two approaches are based on slightly different notions of combinatorial moduli which drive to apparently similar results. The aim of this article is to explain the deep relationships between circle packings and combinatorial moduli of curves, and to compare the approaches to Cannon's conjecture to which they lead. The central result of this article is a theorem that provides a common base to derive, on one hand, a weak version of the combinatorial mapping Riemann theorem of J.~W.~Cannon and, on the other hand, results of M.~Bonk and B.~Kleiner. The crucial point of the demonstration is to use, like M.~Bonk and B.~Kleiner, circle packings in order to translate naturally the combinatorial data to analytic data.
scientific article; zbMATH DE number 5673894
scientific article