Combining Rational maps and Kleinian groups via orbit equivalence

Abstract: We develop a new orbit equivalence framework for holomorphically
combining the dynamics of complex polynomials with that of Kleinian
surface groups. We show that the only torsion-free Fuchsian groups that
can be thus combined 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, 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
combinable in our framework. This is joint work with Sabyasachi Mukherjee.