Complexes of Nonseparating Curves and Mapping Class Groups

Abstract

Let $R$ be a compact, connected, orientable surface of genus $g$, $Mo{d}_R^{\ast }$ be the extended mapping class group of $R$, $\mathcal{C}(R)$ be the complex of curves on $R$, and $\mathcal{N}(R)$ be the complex of nonseparating curves on $R$. We prove that if $g\ge 2$ and $R$ has at most $g-1$ boundary components, then a simplicial map $\lambda :\mathcal{N}(R)\to \mathcal{N}(R)$ is superinjective if and only if it is induced by a homeomorphism of $R$. We prove that if $g\ge 2$ and $R$ is not a closed surface of genus two then $Aut(\mathcal{N}(R))=Mo{d}_R^{\ast }$, and if $R$ is a closed surface of genus two then $Aut(\mathcal{N}(R))=Mo{d}_R^{\ast }/\mathcal{C}(Mo{d}_R^{\ast })$. We also prove that if $g=2$ and $R$ has at most one boundary component, then a simplicial map $\lambda :\mathcal{C}(R)\to \mathcal{C}(R)$ is superinjective if and only if it is induced by a homeomorphism of $R$. As a corollary we prove some new results about injective homomorphisms from finite index subgroups to $Mo{d}_R^{\ast }$. The last two results complete the author's previous results to connected orientable surfaces of genus at least two.

24 pages, 13 figures; The result about automorphism group of complex of nonseparating curves has been extended to compact, connected, orientable surfaces of genus at least two

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