Finite rigid sets in curve complexes

TL;DR

This paper proves that for any finite-type orientable surface, there exists a finite subcomplex of its curve complex such that any locally injective simplicial map from this subcomplex extends to an automorphism of the entire curve complex, demonstrating finite rigidity.

Contribution

The authors establish finite rigidity of curve complexes by identifying a finite subcomplex that determines automorphisms, refining understanding of their symmetries.

Findings

Existence of finite subcomplexes determining automorphisms

Extension of local maps to global automorphisms

Finite rigidity holds for all finite-type orientable surfaces except twice-punctured tori

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 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.