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

We prove that there exists a homeomorphism $\chi$ between the connectedness locus $\mathcal{M}_{\Gamma}$ 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 $\mathcal{M}_{\Gamma}$, 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 $\mathcal{M}_{\Gamma}$ is homeomorphic to $\mathcal{M}$.