On the Arc and Curve Complex of a Surface

We study the 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. AMS Mathematics Subject Classification: Primary 32G15; Secondary