A general dynamical theory of Schwarz reflections, B-involutions, and algebraic correspondences

TL;DR

This paper develops a unified dynamical framework for Schwarz reflections, B-involutions, and algebraic correspondences, demonstrating their matings with polynomials and groups, and exploring their parameter spaces and algebraic structures.

Contribution

It introduces a general theory linking polynomial dynamics, reflection groups, and algebraic correspondences, with new constructions and parameter space analyses.

Findings

Existence of matings between polynomials and reflection groups.

Parameter space slices resemble polynomial map spaces.

Algebraic descriptions of these matings and correspondences.

Abstract

In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with circle maps arising from the corresponding groups. These matings emerge naturally as degenerate (anti-)polynomial-like maps, and we show that the corresponding parameter space slices for such matings bear strong resemblance with parameter spaces of polynomial maps. Furthermore, we provide algebraic descriptions for these matings, and construct algebraic correspondences that combine generic (anti-)polynomials and genus zero orbifolds in a common dynamical plane, providing a new concrete evidence to Fatou's vision of a unified theory of groups and maps.