Bers Slices in Families of Univalent Maps

TL;DR

This paper embeds Bers slices of ideal polygon reflection groups into the family of univalent functions, linking Kleinian groups with anti-holomorphic polynomials and their Julia sets.

Contribution

It constructs a novel embedding of Bers slices into univalent functions and characterizes their images as univalent rational maps, connecting Kleinian groups with anti-holomorphic dynamics.

Findings

Bers slices are embedded into the family of univalent functions.

The image of the embedding is characterized as a family of univalent rational maps.

Limit sets of Kleinian reflection groups are homeomorphic to Julia sets of anti-holomorphic polynomials.

Abstract

We construct embeddings of Bers slices of ideal polygon reflection groups into the classical family of univalent functions $\Sigma$. This embedding is such that the conformal mating of the reflection group with the anti-holomorphic polynomial $z\mapsto\overline{z}^d$ is the Schwarz reflection map arising from the corresponding map in $\Sigma$. We characterize the image of this embedding in $\Sigma$ as a family of univalent rational maps. Moreover, we show that the limit set of every Kleinian reflection group in the closure of the Bers slice is naturally homeomorphic to the Julia set of an anti-holomorphic polynomial.