Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

TL;DR

This paper explores the intersection of rational dynamics and Kleinian groups through algebraic correspondences, revealing unifying structures and applications in their parameter spaces.

Contribution

It provides an overview of the dynamics of algebraic correspondences, focusing on matings between rational maps and Kleinian groups, and highlights their unifying structures.

Findings

Identification of unifying structures in parameter spaces

Connections between moduli spaces of rational maps and Kleinian groups

Applications of developed techniques in algebraic correspondences

Abstract

We present an overview of the rapidly evolving field of dynamics of algebraic correspondences, with a focus on matings between rational maps and Kleinian groups. These correspondences exhibit rich dynamics, both within the Sullivan dictionary and beyond. We highlight unifying structures in their parameter spaces, showing how moduli spaces of rational maps and Kleinian groups naturally connect. We also outline a range of applications of the techniques developed in this framework and conclude with several promising directions.