C*-Algebras Associated With Algebraic Correspondences On The Riemann Sphere

Let p(z,w) be a polynomial in two variables. We call the solution of the algebraic equation p(z,w) = 0 an algebraic correspondence. We regard it as the graph of the multivalued function z → w defined implicitly by p(z,w) = 0. Algebraic correspondences on the Riemann sphere C{double struck}̂ generalize both Kleinian groups and rational functions. We introduce C*-algebras associated with algebraic correspondences on the Riemann sphere. We show that if an algebraic correspondence is free and expansive on a closed p-invariant subset J of C{double-struck}̂, then the associated C*-algebra O_p(J) is simple and purely infinite.