Combinatorial rigidity of the curve complex
Aix-Marseille Université - Site St Charles
3, place Victor Hugo - case 39
13331 MARSEILLE Cedex 03
Given a closed surface S we define the curve complex, C(S), as the simplicial complex whose vertices are the (isotopy classes of essential simple closed) curves in S, and a simplex is a set of pairwise disjoint curves. This complex was introduced by Harvey in 1977 and since then it has been studied due to its relation to several areas of study in geometry and topology.
In this talk we will see different different results concerning the combinatorial rigidity of C(S), that is, finding enough conditions for a simplicial self-map of C(S) to be an automorphism. We will be particularly interested in the concept of finite rigid sets of C(S) and we will give a method to find a sequence of (increasing) finite rigid sets whose union is C(S). Finally, we will give several consequence of this method concerning self-maps of C(S), self-maps of other simplicial graphs of multicurves of S, and homomorphisms of certain subgroups of the mapping class group of S.