# Schwarz reflections and anti-holomorphic correspondences

## TL;DR

This paper explores the dynamics of Schwarz reflections, transforming them into parabolic rational maps, and establishes a mating framework with anti-holomorphic correspondences related to the modular group.

## Contribution

It introduces a quasiconformal surgery to relate Schwarz reflections to parabolic rational maps and develops a mating theory with anti-holomorphic correspondences.

## Key findings

- Straightening map between Schwarz reflection parameter plane and parabolic Tricorn
- Schwarz reflections produce anti-holomorphic correspondences as matings with the modular group
- Comparison with classical Bullett-Penrose family of matings

## Abstract

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in \cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic anti-rational maps with the abstract modular group. We further illustrate our mating framework by studying the correspondence associated with the Schwarz reflection map of a deltoid.