Finite rigid sets in curve complexes

Abstract

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 X -> C(S) is the restriction of an element of Aut(C(S)), unique up to the (finite) pointwise 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 Mod(+/-)(S).