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 λ : 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 authorʹs previous results about closed connected orientable surfaces of genus at least 3, to the surface R.