Mahan Mj

An orbit equivalence framework for combining rational maps and Kleinian groups

We discuss a recent orbit equivalence framework for holomorphically mating the dynamics of complex polynomials with that of Kleinian surface groups. We start by demanding a strict orbit equivalence and show that the only torsion-free Fuchsian groups that can be thus mated are punctured sphere groups. We then relax the strict orbit equivalence framework to a virtual orbit equivalence framework allowing us to include certain special orbifolds including the modular surface. Finally, a further relaxation of this framework demanding only orbit equivalence for generic points allows us to expand the scope considerably to all orbifolds of arbitrary genus excluding only compact orbifolds that do not admit an essential simple separating curve, i.e. (p,q,r) orbifolds. Time-permitting we shall mention a closely related framework of algebraic correspondences that arises out of these matings addressing an old question of Fatou.

This is joint work with Yusheng Luo and Sabyasachi Mukherjee.