Let’s get into the mapping class group and talk a bit about its elements and its structure. I’m going to omit proofs, because I can’t beat Minsky’s exposition, and this is just some flavor and definitions, most of which won’t be coming up too much in the future.
So, on a torus, we classified elements of the mapping class group. Elliptic elements will remain the things of finite order, so we’re going to focus on parabolic elements, which become Dehn twists, and hyperbolic elements that become pseudo-Anosov maps.
Definition: A Dehn twist is a map that fixes the surface outside of an annulus on the annulus lifts to on . The isotopy class of this map only depends on the isotopy class of the embedding of the annulus into the surface.
A Dehn twist is particularly simple on homology, it takes a class to where is the class being twisted and is the intersection number. Denote the Dehn twist around a curve by .
Dehn twists are absolutely fundamental, in fact:
Theorem: The mapping class group of a surface is finitely presented, and the subgroup that does not permute the punctures is generated by finitely many Dehn twists on nonseparating curves.
Furthermore, the relations in the subgroup generated by the Dehn twists are generated by:
- Dehn twists on disjoint curves commute
- If and intersect exactly once, then , the Braid relation
- If is a separating curve that cuts off a torus, and are curves in the torus that intersect exactly once, then .
- The Lantern Relation
Unfortunately, it would take me another post or two to properly define everything needed to say what a pseudo-Anosov map is, but here’s the definition:
Definition: A map is pseudo-Anosov is it is isotopic to a map with a pair of measured foliations that can be realized transversely and with the same singular points, so that preserves each one and multiplies the transverse measure on by and by .
On the other hand, I can appeal to a big theorem that I also won’t prove!
Nielsen-Thurston Classification Theorem: For every , one of the following holds:
- has finite order
- There is a system of disjoint essential simple closed curves fixed (up to isotopy) by
- has a pseudo-Anosov representative.
The first two aren’t mutually exclusive, but anything that isn’t one of them must be pseudo-Anosov, and I’ll just leave it at that, and move on. Next time, back to Teichmüller space.