Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups

Abstract

Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mo{d}_S^{\ast }$ be the extended mapping class group of $S$. We prove that a simplicial map, $\lambda :\mathcal{C}(S)\to \mathcal{C}(S)$, preserves nondisjointness (i.e. if $\alpha$ and $\beta$ are two vertices in $\mathcal{C}(S)$ and $i(\alpha ,\beta )\ne 0$, then $i(\lambda (\alpha ),\lambda (\beta ))\ne 0$) iff it is induced by a homeomorphism of $S$. As a corollary, we prove that if $K$ is a finite index subgroup of $Mo{d}_S^{\ast }$ and $f:K\to Mo{d}_S^{\ast }$ is an injective homomorphism, then $f$ is induced by a homeomorphism of $S$ and $f$ has a unique extension to an automorphism of $Mo{d}_S^{\ast }$.

34 pages, 17 figures