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

Abstract

In this article, 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.

On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps

“…In the recent work [LLM20], the authors constructed an explicit dynamically meaningful correspondence between critically fixed anti-holomorphic rational maps and kissing Kleinian reflection groups acting on the Riemann sphere. The aim of the current paper is to investigate some fundamental parameter space implications of the above correspondence; specifically, to compare boundedness properties and mutual interaction structure of critically fixed hyperbolic components and deformation spaces of kissing reflection groups.…”

“…Before we proceed to give the precise statements of our principal results, let us summarize the dynamical correspondence established in [LLM20].…”

“…A graph is called k-connected if it contains more than k vertices and remains connected with any k − 1 vertices removed. In [LLM20], we showed that the planar dual Γ of T is 2-connected and simple with d + 1 vertices. Conversely, given a 2-connected, simple, plane graph Γ with d + 1 vertices, the planar dual of Γ is isomorphic to the Tischler graph of a degree d critically fixed anti-rational map R Γ , which is unique up to Möbius conjugation (cf.…”

“…In consistence with the convention adopted in [LLM20], a plane graph will mean a planar graph together with an embedding in C throughout this paper. We say that two plane graphs are isomorphic if the underlying graph isomorphism is induced by an orientation preserving homeomorphism of C. Two kissing reflection groups with isomorphic contact graphs are quasiconformally conjugate.…”

“…Thus, by the circle packing theorem, the quasiconformal conjugacy classes of kissing reflection groups are in correspondence with isomorphism classes of simple plane graphs. It can be verified that the limit set of G = G Γ is connected if and only if the corresponding contact graph Γ is 2-connected (see [LLM20,§3]).…”