  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.
