On the arc and curve complex of a surface

Mustafa Korkmaz (1) , Athanase Papadopoulos (2, 3)

1 METU - Department of Mathematics (METU)
2 IRMA - Institut de Recherche Mathématique Avancée
3 MPI - Max-Planck-Institut für Mathematik

We study the {\it arc and curve} complex $AC(S)$ of an oriented connected surface $S$ of finite type with punctures. We show that except for some finitely many special surfaces, the simplicial automorphism group of $AC(S)$ coincides with the natural image of the extended mapping class group of $S$ in that group. We also show that for any vertex of $AC(S)$, the combinatorial structure of the link of that vertex is sufficient to characterize the topological type of the curve or of the arc on $S$ that represents this vertex. Finally, we show that the natural embedding of the curve complex in $AC(S)$ is a quasi-isometry.

Keywords: mapping class group.; curve complex; arc complex; arc and curve complex; mapping class group

Domains: Geometric Topology [math.GT]

Mustafa Korkmaz, Athanase Papadopoulos. On the arc and curve complex of a surface. 2009. ⟨hal-00405188v1⟩