Complexes of nonseparating curves and mapping class groups

By Elmas Irmak. 2004

70 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f21894a6a196213ddb9f6a00082a0ef7134b0c72

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Injective simplicial maps of the arc complex (2006, background): “metric objects on surfaces. Some of these objects include Schaller’s complex (see [S] by Schaller), the complex of pants decompositions (see [M] by Margalit), the complex of nonseparating curves (see [Ir3] by Irmak), the complex of separating curves (see [BM1] by Brendle-Margalit, and [MV] by McCart”
Curve Complexes and Finite Index Subgroups of Mapping Class Groups (2005, methodology): “and yes for all other Sg,n with g � 2 [17] [16] [ 15 ].”