Circle packings, kissing reflection groups and critically fixed anti-rational maps

Abstract

In this paper, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.

49 pages, 14 figures. The last two sections of v1 are moved to the sequel of this paper. Final version, to appear in Forum of Math. Sigma

References in corpus (5)

• Bers Slices in Families of Univalent Maps
• Sharpness of connectivity bounds for quadrature domains
• Bounded hyperbolic components of bicritical rational maps
• On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps
• Classification of critically fixed anti-Thurston maps

Cited by in corpus (5)

• Combining Rational maps and Kleinian groups via orbit equivalence
• On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps
• The Sullivan dictionary and Bowen-Series maps
• On quasiconformal non-equivalence of gasket Julia sets and limit sets
• Combination Theorems in Groups, Geometry and Dynamics