Factorization rules in quantum Teichmüller theory

For a punctured surface S , a point of its Teichmüller space T .S/ determines an irreducible representation of its quantization T q .S /.We analyze the behavior of these representations as one goes to infinity in T .S /, or in the moduli space M.S / of the surface.The main result of this paper states that an irreducible representation of T q .S / limits to a direct sum of representations of T q .S /, where S is obtained from S by pinching a multicurve to a set of nodes.The result is analogous to the factorization rule found in conformal field theory.