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

TL;DR

This paper establishes a direct link between kissing reflection groups and critically fixed anti-rational maps using planar graphs, leading to new insights into their dynamics and solutions to geometric mating problems.

Contribution

It introduces an explicit correspondence between these groups and maps, enabling new analysis of their dynamical properties and solving open problems in geometric mating.

Findings

Established a correspondence using planar graphs

Provided complete solutions to geometric mating problems

Analyzed dynamical consequences of the correspondence

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.