Continuity of matings of Kleinian groups and polynomials

TL;DR

This paper explores the mating of Kleinian groups and polynomials through a surgical construction, extending previous methods to higher degrees and establishing regularity properties of the resulting maps.

Contribution

It advances the theory of holomorphic correspondences by generalizing the mating construction and proving key regularity results on parameter spaces.

Findings

Proved analyticity of the mating map on its domain interior.

Established continuity of the map under quasiconformal rigidity on the boundary.

Extended the surgical construction to higher degree maps.

Abstract

In recent years, the study of holomorphic correspondences as dynamical systems that can display behaviors of both rational maps and Kleinian groups has gained a good amount of attention. This phenomenon is related to the Sullivan dictionary, a list of parallels between the theories of these two systems. We build upon a surgical construction of such matings, due to Bullett and Harvey, increasing the degree of maps we consider, and proving regularity properties of the mating map on parameter spaces: namely, analyticity on the interior of its domain of definition, and continuity under quasiconformal rigidity on the boundary.