It was envisaged by Fatou in the 1920s that dynamics of rational maps and actions of Kleinian groups on the Riemann sphere can be studied in the common framework of iterated algebraic correspondences. In a pioneering work in the 1990s, Bullett and Penrose produced first examples of algebraic correspondences that arise as combinations/matings of quadratic polynomials with the modular group.
The goal of this talk is to outline a general theory of such correspondences. We will discuss the existence of algebraic correspondences that arise as matings of large classes of Fuchsian groups (including punctured sphere groups and modular/Hecke groups) with generic polynomials having connected Julia set. The construction will reveal interesting connections between correspondences, pinched polynomial-like maps, and a new class of holomorphic dynamical systems generated by "B-involutions".