On the arc and curve complex of a surface

Source abstract

Abstract We study the arc and curve complex AC ( S ) of an oriented connected surface S of finite type with punctures. We show that if the surface is not a sphere with one, two or three punctures nor a torus with one puncture, then 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 characterizes the type of a curve or of an arc in S that represents that vertex. We also give a proof of the fact if S is not a sphere with at most three punctures, then the natural embedding of the curve complex of S in AC ( S ) is a quasi-isometry. The last result, at least under some slightly more restrictive conditions on S , was already known. As a corollary, AC ( S ) is Gromov-hyperbolic.