Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups

Let S be a closed, connected, orientable surface of genus at least 3, C(S) be the complex of curves on S and ModS∗ be the extended mapping class group of S. We prove that a simplicial map, λ : C(S)→C(S), preserves nondisjointness (i.e. if α and β are two vertices in C(S) and i(α,β)≠0, then i(λ(α),λ(β))≠0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of ModS∗ and f : K→ModS∗ is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of ModS∗.