Curve complex

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.

Curve complexes

Definition

Let $S$ be a finite type connected oriented surface. More specifically, let ${\displaystyle S=S_{g,b,n}}$ be a connected oriented surface of genus ${\displaystyle g\geq 0}$ with ${\displaystyle b\geq 0}$ boundary components and $n\geq 0$ punctures.

The curve complex ${\displaystyle C(S)}$ is the simplicial complex defined as follows:^[1]

• The vertices are the free homotopy classes of essential (neither homotopically trivial nor peripheral) simple closed curves on $S$;
• If $c_{1},\ldots ,c_{n}$ represent distinct vertices of ${\displaystyle C(S)}$, they span a simplex if and only if they can be homotoped to be pairwise disjoint.

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 $S$ is homeomorphic to a torus or a punctured torus the resulting complex complex is a 2-complex with 1-skeleton isomorphic to the Farey graph.

Geometry of the curve complex

Basic properties

If $S$ is a compact surface of genus $g>1$ with $b$ boundary components the dimension of ${\displaystyle C(S)}$ is equal to ${\displaystyle \xi (S)=3g-3+b}$. The complex of curves is never locally finite (each point belongs to infinitely many top-dimensional simplices), in fact it is homotopically equivalent to a wedge sum of infinitely many spheres

Intersection numbers and distance on C(S)

The combinatorial distance on the 1-skeleton of ${\displaystyle C(S)}$ 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^[2]

${\displaystyle d_{S}(\alpha ,\beta )\leq 2\log _{2}(i(\alpha ,\beta ))+2}$

for any two nondisjoint simple closed curves $\alpha ,\beta$. 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.^[3]

Hyperbolicity

It was proved by Masur and Minsky^[4] 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.^[3][5]

Relation with the mapping class group and Teichmüller space

Action of the mapping class group

The mapping class group of $S$ acts on the complex ${\displaystyle C(S)}$ in the natural way: it acts on the vertices by ${\displaystyle \phi \cdot \alpha =\phi _{*}\alpha }$ and this extends to an action on the full complex. This action allows to prove many interesting properties of the mapping class groups.^[6]

While the mapping class group itself is not an hyperbolic group, the fact that ${\displaystyle C(S)}$ is hyperbolic still has implications for its structure and geometry.^[7][8]

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.

Applications to 3-dimensional topology

Heegard splittings

A simplex in ${\displaystyle C(S)}$ determines a "filling" of $S$ to a handlebody. Choosing two simplices in ${\displaystyle C(S)}$ thus determines a Heegard splitting of a three-manifold,^[9] with the additional data of an Heegard diagram (a maximal system of disjoint simple closed curves bounding disks for each of the two handlebodies). Some properties of Heegard 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 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 (in the sense of the geometrisation conjecture of the manifold and vice versa.^[9] A guiding principle is that the minimal distance of a Heegard splitting is a measure of the complexity of the manifold.^[10]

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.^[11] 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.^[12] The proof that this model is hyperbolic almost surely (in a certain sense) uses the geometry of the complex of curves.^[13]

Notes

1. Farb and Margalit, Ch. 4.1, p. 92
2. Schleimer 2006, Lemma 1.21.
3. Bowditch 2006.
4. Masur & Minsky 1999.
5. Aougab, Tarik (2013). "Uniform hyperbolicity of the graphs of curves". Geom. Topol. 17: 2855–2875. doi:10.2140/gt.2013.17.2855. MR 3190300.
6. Ivanov 1992, Chapter 7.
7. Manganas, Johanna (2010). "Uniform uniform exponential growth of subgroups of the mapping class group". Geom. Funct. Anal. 19: 1468–1480. MR 2585580.
8. Dahmani, François; Guirardel, Vincent; Osin, Denis. "Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces".
9. Hempel 2001.
10. Abrams, Aaron; Schleimer, Saul (2005). "Distances of Heegaard splittings". Geom. Topol. 9: 95–119. doi:10.2140/gt.2005.9.95. MR 2115669.
11. Bowditch, Brian H. (2005). "Hyperbolic 3-manifolds and the geometry of the curve complex". European Congress of Mathematics. Eur. Math. Soc. pp. 103–115.
12. Dunfield, Nathan M.; Thurston, William P. (2006). "Finite covers of random 3-manifolds". Invent. Math. 166: 457–521. doi:10.1007/s00222-006-0001-6. MR 2257389.
13. Maher, Joseph (2010). "Random Heegaard splittings". J. Topol. 3: 997–1025. doi:10.1112/jtopol/jtq031.

References

• Bowditch, Brian H. (2006). "Intersection numbers and the hyperbolicity of the curve complex". J. Reine Angew. Math. 598: 105–129. MR 2270568.
• Hempel, John (2001). "3-manifolds as viewed from the curve complex". Topology. 40: 631–657. doi:10.1016/s0040-9383(00)00033-1. MR 1838999.
• Ivanov, Nikolai (1992). Subgroups of Teichmüller Modular Groups. American Math. Soc.
• Masur, Howard A.; Minsky, Yair N. (1999). "Geometry of the complex of curves. I. Hyperbolicity". Invent. Math. 138: 103–149. doi:10.1007/s002220050343. MR 1714338.
• Schleimer, Saul (2006). "Notes on the complex of curves" (PDF).
• Benson Farb and Dan Margalit, A primer on mapping class groups. Princeton Mathematical Series, 49. Princeton University Press, Princeton, NJ, 2012. ISBN 978-0-691-14794-9