  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 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.
