Abstract of E. Irmak's Thesis

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Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups, Ph. D. Thesis, Michigan State University, 2002

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