Algebraic correspondences: Where rational dynamics meets Kleinian groups

par Luna Lomonaco

vendredi 9 oct. 2026, 10:30 → 12:30 Europe/Paris

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 $F_f$
and Julia set $J_f$ of a holomorphic map $f$ on the Riemann sphere
$\widehat{\mathbb{C}}$, and the ordinary set $\Omega(G)$ and limit set
$\Lambda(G)$ of a finitely generated Kleinian group $G$ acting on
$\widehat{\mathbb{C}}$. 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 $\widehat{\mathbb{C}}$ is a
multivalued map $\mathcal{F}: z \mapsto w$ 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 \Omega, 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.