Algebraic correspondences: Where rational dynamics meets Kleinian groups
Luna Lomonaco (IMPA)

The analogies between the iteration of holomorphic maps and the action of Kleinian groups were first systematically explored by Dennis Sullivan in the mid-1980s. In the landmark paper, where he famously proved Fatou’s conjecture—that rational maps on the Riemann sphere have no wandering domains—Sullivan introduced what is now known as Sullivan’s Dictionary. This conceptual framework draws deep parallels between the definitions, theorems, and conjectures of holomorphic dynamics and those of Kleinian group theory. Sullivan emphasized striking similarities between the Fatou set Ff and Julia set Jf of a holomorphic map f on the Riemann sphere bC , and the ordinary set (G) and limit set (G) of a finitely generated Kleinian group G acting on bC . His proof of the no wandering domains theorem was directly inspired by methods used to establish Ahlfors’ Finiteness Theorem in the setting of Kleinian groups, highlighting the profound conceptual bridges between the two fields. Both rational maps and finitely generated Kleinian groups can be regarded as special cases of holomorphic correspondences. An n-to-m holomorphic correspondence on bC is a multivalued map F : z 7! w defined implicitly by a polynomial relation P(z;w) = 0. In 1994, Shaun Bullett and Christopher Penrose introduced the first family of correspondences that contains matings between quadratic rational maps and the modular group, and proved that, for a particular parameter, the correspondence is a mating: it behaves as the modular group on an open subset , and as a polynomial (and its inverse) in the complement. Since then, the field of correspondences which are matings between rational maps and Kleinian groups grew considerably. In this talk I will give an overview of the subject..