Complexes of Nonseparating Curves and Mapping Class Groups

TL;DR

This paper characterizes automorphisms and superinjective maps of complexes of curves and nonseparating curves on surfaces, linking them to homeomorphisms and the extended mapping class group, with new results for genus two surfaces.

Contribution

It establishes that superinjective maps of these complexes are induced by homeomorphisms and determines automorphism groups for various surface types, extending previous work.

Findings

Superinjective maps of nonseparating curve complexes are induced by homeomorphisms.

Automorphism groups of the complexes are identified with the extended mapping class group.

New results on injective homomorphisms from finite index subgroups to the mapping class group.

Abstract

Let $R$ be a compact, connected, orientable surface of genus $g$, $Mod_R^*$ 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 $g \geq 2$ 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 $g \geq 2$ and $R$ is not a closed surface of genus two then $Aut(\mathcal{N}(R))= Mod_R^*$, and if $R$ is a closed surface of genus two then $Aut(\mathcal{N}(R))= Mod_R ^* /\mathcal{C}(Mod_R^*)$. 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 $Mod_R^*$. The last two results complete the author's previous results to connected orientable surfaces of genus at least two.