We develop a new orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers’ boundary groups that are mateable in our framework. Broadly speaking, the first talk (by Sabyasachi Mukherjee) will emphasize the aspects to do with complex dynamics more, and the second (by Mahan Mj) will lay emphasis on the 1-dimensional dynamics aspects of the problem.