  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.
