Correlation Functions for Some Conformal Theories on Riemann Surfaces

We discuss the geometrical connection between 2-D conformal field theories, random walks on hyperbolic Riemann surfaces and knot theory. For the wide class of CFTs with monodromies being the discrete subgroups of SL(2, &R;), the determination of four-point correlation functions are related to construction of topological invariants for random walks on multipunctured Riemann surfaces.