Curve complex

Summary

The curve complex of a surface is a simplicial complex. Its vertices are the isotopy classes of essential simple closed curves, and its simplices are sets of curves that can be made pairwise disjoint. The mapping class group acts on it by simplicial automorphisms. Harvey introduced it, and Masur and Minsky proved that it is hyperbolic in the sense of Gromov. It is a central tool in the study of mapping class groups and of Teichmüller space.