Mating quadratic maps with the modular group II

Abstract

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences ${\mathcal{F}}_a$: $${\left(\frac{aw-1}{w-1}\right)}^2+\left(\frac{aw-1}{w-1}\right)\left(\frac{az+1}{z+1}\right)+{\left(\frac{az+1}{z+1}\right)}^2=3$$ and proved that for every value of $a\in [4,7]\subset \mathbb{R}$ the correspondence ${\mathcal{F}}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,c\in \mathbb{R}$ and the modular group $\Gamma =PSL(2,\mathbb{Z})$. They conjectured that this is the case for every member of the family ${\mathcal{F}}_a$ which has $a$ in the connectedness locus. We prove here that every member of the family ${\mathcal{F}}_a$ which has $a$ in the connectedness locus is a mating between the modular group and an element of the parabolic quadratic family $Pe{r}_1(1)$.

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