In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups.
be a finite type connected oriented surface.
is the simplicial complex defined as follows:[1] For surfaces of small complexity (essentially the torus, punctured torus, and four-holed sphere), with the definition above the curve complex has infinitely many connected components.
One can give an alternate and more useful definition by joining vertices if the corresponding curves have minimal intersection number.
With this alternate definition, the resulting complex is isomorphic to the Farey graph.
The complex of curves is never locally finite (i.e. every vertex has infinitely many neighbors).
is in fact homotopically equivalent to a wedge sum of spheres.
is related to the intersection number between simple closed curves on a surface, which is the smallest number of intersections of two curves in the isotopy classes.
For example[3] for any two nondisjoint simple closed curves
One can compare in the other direction but the results are much more subtle (for example there is no uniform lower bound even for a given surface) and harder to prove.
[4] It was proved by Masur and Minsky[5] that the complex of curves is a Gromov hyperbolic space.
Later work by various authors gave alternate proofs of this fact and better information on the hyperbolicity.
and this extends to an action on the full complex.
This action allows to prove many interesting properties of the mapping class groups.
[8][9] There is a natural map from Teichmüller space to the curve complex, which takes a marked hyperbolic structures to the collection of closed curves realising the smallest possible length (the systole).
It allows to read off certain geometric properties of the latter, in particular it explains the empirical fact that while Teichmüller space itself is not hyperbolic it retains certain features of hyperbolicity.
thus determines a Heegaard splitting of a three-manifold,[10] with the additional data of an Heegaard diagram (a maximal system of disjoint simple closed curves bounding disks for each of the two handlebodies).
Some properties of Heegaard splittings can be read very efficiently off the relative positions of the simplices: In general the minimal distance between simplices representing diagram for the splitting can give information on the topology and geometry (in the sense of the geometrisation conjecture of the manifold) and vice versa.
[10] A guiding principle is that the minimal distance of a Heegaard splitting is a measure of the complexity of the manifold.
[11] As a special case of the philosophy of the previous paragraph, the geometry of the curve complex is an important tool to link combinatorial and geometric properties of hyperbolic 3-manifolds, and hence it is a useful tool in the study of Kleinian groups.
[15] The proof that this model is hyperbolic almost surely (in a certain sense) uses the geometry of the complex of curves.