Automorphisms of complexes of curves and of Teichm uller spaces

To every compact orientable surface one can associate following Harvey Ha Ha a combinatorial object the so called complex of curves which is analogous to Tits buildings associated to semisimple Lie groups The basic result of the present paper is an analogue of a fundamental theorem of Tits for these complexes It asserts that every automorphism of the complex of curves of a surface is induced by some element of the Teichm uller modular group of this surface or what is the same by some di eomorphism of the surface in question This theorem allows us to give a completely new proof of a famous theorem of Royden R about isometries of the Teichm uller space In contrast with Royden s proof which is local and analytic this new proof is a global and geometric one and reveals a deep analogy between Royden s theorem and the Mostow s rigidity theorem Mo Mo Another application of our basic theorem is a complete description of isomorphisms between subgroups of nite index of a Teichm uller modular group This result in its turn has some further applications to modular groups