We will discuss a construction of algebraic correspondences on compact Riemann surfaces of arbitrary genus that combine actions of rational maps and Kleinian groups. This involves manufacturing appropriate meromorphic maps on subsets of the sphere via surgery/tuning techniques and characterizing the resulting meromorphic maps as algebraic functions.

At a parameter space level, this gives rise to 'products' of Teichmüller spaces of genus zero orbifolds and connectedness loci of polynomials inside appropriate Hurwitz spaces (i.e., moduli spaces of ramified covers of the Riemann sphere). Time permitting, we will mention some boundedness properties of such embeddings of Teichmüller spaces and connectedness loci.