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

We prove that there exists a homeomorphism χ between the connectedness locus $M_{\Gamma }$ for the family $F_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $M_1$. The homeomorphism χ is dynamical ($F_a$ is a mating between $PSL(2,Z)$ and $P_{\chi \left(a\right)}$), it is conformal on the interior of $M_{\Gamma }$, 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 $M_1$ is homeomorphic to the classical Mandelbrot set $M$, we deduce that $M_{\Gamma }$ is homeomorphic to $M$.