The quantum Teichmller space of a punctured surface is a purely combinatorial/algebraic object associated to this surface. We investigate a certain type of representations of the corresponding algebra, called local representations. These are essentially classified by group homomorphisms from the fundamental group of the surface to the isometry group of hyperbolic 3-space. Using hyperbolic metrics on 3-manifolds fibering over the circle, we use this construction to extract invariants of diffeomorphisms of surfaces.