  We study the arc complex of a surface with marked points in the interior and
on the boundary. We prove that the isomorphism type of the arc complex
determines the topology of the underlying surface, and that in all but a few
cases every automorphism is induced by a homeomorphism of the surface. As an
application we deduce some rigidity results for the Fomin-Shapiro-Thurston
cluster algebra associated to such a surface. Our proofs do not employ any
known simplicial rigidity result.
