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

Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod*(R) be the extended mapping class group of R. Suppose that either g=2 and p >= 2 or g >= 3 and p >= 0. We prove that a simplicial map lambda:C(R)-> C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod*(R) and f:K -> Mod*(R) is an injective homomorphism, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod*(R). This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R. (C) 2005 Elsevier B.V. All rights reserved.