Combining Rational maps and Kleinian groups via orbit equivalence

Abstract: 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 Teichm\"uller space of matings has one component corresponding to Bowen-Series maps and one corresponding to higher Bowen-Series maps. We 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.