Superinjective Simplicial Maps of Complexes of Curves and Injective Homomorphisms of Subgroups of Mapping Class Groups
Abstract
Description
Let $S$ be a closed, connected, orientable surface of genus at least 3, $\mathcal{C}(S)$ be the complex of curves on $S$ and $Mod_S^*$ be the extended mapping class group of $S$. We prove that a simplicial map, $λ: \mathcal{C}(S) \to \mathcal{C}(S)$, preserves nondisjointness (i.e. if $α$ and $β$ are two vertices in $\mathcal{C}(S)$ and $i(α, β) \neq 0$, then $i(λ(α), λ(β)) \neq 0$) 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 $f: K \to Mod_S^*$ is an injective homomorphism, then $f$ is induced by a homeomorphism of $S$ and $f$ has a unique extension to an automorphism of $Mod_S^*$.
34 pages, 17 figures
34 pages, 17 figures