Mating parabolic rational maps with Hecke groups

TL;DR

This paper proves that certain degree d rational maps with parabolic fixed points can be mated with Hecke groups, using algebraic correspondences, confirming a conjecture about their mateability.

Contribution

It establishes the mateability of specific parabolic rational maps with Hecke groups through a novel construction of algebraic correspondences.

Findings

Rational maps with parabolic fixed points are mateable with Hecke groups.

Construction of a pinched polynomial-like map as a key step.

Lifting the map to an algebraic correspondence confirms the conjecture.

Abstract

We prove that any degree $d$ rational map having a parabolic fixed point of multiplier $1$ with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group $H_{d+1}$, with the mating realized by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made in \cite{BF1}. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering.