ar X iv : m at h / 04 07 08 6 v 1 [ m at h . G T ] 6 J ul 2 00 4 REPRESENTATIONS OF THE QUANTUM TEICHMÜLLER SPACE AND INVARIANTS OF SURFACE DIFFEOMORPHISMS

— We investigate the representation theory of the polynomial core T S of the quantum Teichmüller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of T S are classified, up to finitely many choices, by group homomorphisms from the fundamental group π1(S) to the isometry group of the hyperbolic 3–space H. We exploit this connection between algebra and hyperbolic geometry to exhibit invariants of diffeomorphisms of S.