Finite rigid sets in curve complexes

We prove that curve complexes of surfaces are finitely rigid: for every orientable surface S of finite topological type, we identify a finite subcomplex X of the curve complex C(S) such that every locally injective simplicial map from X into C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) point-wise stabilizer of X in Aut(C(S)). Furthermore, if S is not a twice-punctured torus, then we can replace Aut(C(S)) in this statement with the extended mapping class group.Comment: 19 pages, 12 figures. v2: small additions to improve exposition. v3: conclusion of Lemma 2.5 weakened, and proof of Theorem 3.1 adjusted accordingly. Main theorem remains unchange