Curve complex


In mathematics, the curve complex is a simplicial complex C 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. It was introduced by W.J.Harvey in 1978.

Curve complexes

Definition

Let be a finite type connected oriented surface. More specifically, let be a connected oriented surface of genus with boundary components and punctures.
The curve complex is the simplicial complex defined as follows:
For surfaces of small complexity, 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.

Geometry of the curve complex

Basic properties

If is a compact surface of genus with boundary components the dimension of is equal to. In what follows, we will assume that. The complex of curves is never locally finite. A result of Harer asserts that is in fact homotopically equivalent to a wedge sum of spheres.

Intersection numbers and distance on ''C''(''S'')

The combinatorial distance on the 1-skeleton of 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
for any two nondisjoint simple closed curves. One can compare in the other direction but the results are much more subtle and harder to prove.

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.

Relation with the mapping class group and Teichmüller space

Action of the mapping class group

The mapping class group of acts on the complex in the natural way: it acts on the vertices by and this extends to an action on the full complex. This action allows to prove many interesting properties of the mapping class groups.
While the mapping class group itself is not a hyperbolic group, the fact that is hyperbolic still has implications for its structure and geometry.

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. 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.

Applications to 3-dimensional topology

Heegaard splittings

A simplex in determines a "filling" of to a handlebody. Choosing two simplices in thus determines a Heegaard splitting of a three-manifold, with the additional data of an Heegaard diagram. Some properties of Heegaard splittings can be read very efficiently off the relative positions of the simplices:
  • the splitting is reducible if and only if it has a diagram represented by simplices which have a common vertex;
  • the splitting is weakly reducible if and only if it has a diagram represented by simplices which are linked by an edge.
In general the minimal distance between simplices representing diagram for the splitting can give information on the topology and geometry and vice versa. A guiding principle is that the minimal distance of a Heegaard 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 properties 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 Heegaard splittings. The proof that this model is hyperbolic almost surely uses the geometry of the complex of curves.