Automorphisms of complexes of curves on punctured spheres and on punctured tori

Let S be either a sphere with ≥5 punctures or a torus with ≥3 punctures. We prove that the automorphism group of the complex of curves of S is isomorphic to the extended mapping class group M∗S. As applications we prove that surfaces of genus ≤1 are determined by their complexes of curves, and any isomorphism between two subgroups of M∗S of finite index is the restriction of an inner automorphism of M∗S. We conclude that the outer automorphism group of a finite index subgroup of M∗S is finite, extending the fact that the outer automorphism group of M∗S is finite. For surfaces of genus ≥2, corresponding results were proved by Ivanov (IHES/M/89/60, Preprint).