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.
Contents
Definition
Let
The curve complex
Examples
For surfaces of small complexity (essentially the torus, punctured torus, and three-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. In case where
Basic properties
If
Intersection numbers and distance on C(S)
The combinatorial distance on the 1-skeleton of
for any two nondisjoint simple closed curves
Hyperbolicity
It was proved by Masur and Minsky 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.
Action of the mapping class group
The mapping class group of
While the mapping class group itself is not an hyperbolic group, the fact that
Comparison with Teichmüller space
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.
Heegard splittings
A simplex in
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. A guiding principle is that the minimal distance of a Heegard splitting is a measure of the complexity of the manifold.
Kleinian groups
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 propertoes of hyperbolic 3-manifolds, and hence it is a useful tool in the study of Kleinian groups. For example, it has been used in the proof of the ending lamination conjecture.
Random manifolds
A possible model for random 3-manifolds is to take random Heegard splittings. The proof that this model is hyperbolic almost surely (in a certain sense) uses the geometry of the complex of curves.