Mating quadratic maps with the modular group III: The modular Mandelbrot set

We prove that there exists a homeomorphism $\chi$ between the connectedness locus ${\mathcal{M}}_{\mathrm{\Gamma }}$ for the family ${\mathcal{F}}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set ${\mathcal{M}}_1$. The homeomorphism $\chi$ is dynamical (${\mathcal{F}}_a$ is a mating between $PSL(2,\mathbb{Z})$ and $P_{\chi (a)}$), it is conformal on the interior of ${\mathcal{M}}_{\mathrm{\Gamma }}$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that ${\mathcal{M}}_1$ is homeomorphic to the classical Mandelbrot set $\mathcal{M}$, we deduce that ${\mathcal{M}}_{\mathrm{\Gamma }}$ is homeomorphic to $\mathcal{M}$.