Quasi-Fuchsian correspondences.

M.PhilWe consider the action of holomorphic correspondences or equally algebraic functions acting on the Riemann sphere C and their limit sets: a holomorphic correspondence is a polynomial relation, P(z;w) = 0 in z and w: A holomorphic correspondence P(z;w) = 0 is said to be an (n : m) correspondence if the degrees of z and w in P are n and m respectively. We identify a class of (2 : 2) holomorphic correspondences whose limit set is a topological circle where on a component of the complement of the limit set, the action of the correspondence is conjugate to the action of the Modular group PSL(2; Z) on the upper half plane. Further, we generalise these results to a class of (3 : 3) holomorphic correspondences with analogous properties