The modular group $PSL_2(\mathbb Z)$ is rigid as a Kleinian group. However it has a space of deformations when it is regarded as a $(2:2)$ correspondence $z \to w$ (defined by the polynomial relation $(w-(z+1))(w(z+1)-z)=0$). We examine a slice of deformation space analogous to the Bers slice for Kleinian groups, the critical relations that occur at isolated interior points of the slice, and the degeneracies that occur at the boundary - one example is a continuous deformation of the action of the modular group on the upper half plane into the action of the polynomial $z \to z^2+1/4$ on its filled Julia set.