Welcome back, and hope all you readers had a good 2014 and particularly good holidays and new year’s celebrations, if you do those things. Today, we’re going to keep on the road to producing the moduli space of curves, by nailing down some more hyperbolic geometry.
Secretly knowing the answer to what the dimension of the moduli space is (over the reals, it’s ), I know how many coordinates we need to construct. Half of them come immediately from the discussion last time of pairs of pants:
Theorem: For any triple of positive numbers , there exists a unique, up to isometry, hyperbolic pair of pants with boundary circles of length , and .
This is actually not a hard theorem to prove. It suffices to show that there is a unique, right angled, convex hyperbolic hexagon with three, non-pairwise-consecutive sides of lengths . Then take another copy and glue them together along the other sides. The existence and uniqueness of such hexagons is a pretty straightforward construction in hyperbolic geometry, just like similar things in Euclidean geometry are.
Now, we fix a pair of pants decomposition of a hyperbolic surface. Last time, we said that you need curves to define it, and their lengths provide us with coordinates , but that’s not quite enough to reconstruct the hyperbolic surface completely.
To try to build the surface, we’ve currently just got a pile of pairs of pants, we need to figure out how to glue them together. We could just identify them by setting the seams (remember that we got them by gluing together hexagons) to be a single curve, but that’s not the only way. We can still rotate them with respect to each other, and we get isometric surfaces every time the rotation parameter changes by .
However, we don’t want to try to directly parameterize hyperbolic structures, we actually need more data. A marking of a hyperbolic surface is a homeomorphism preserving orientation from a topological surface , and a pair of marked hyperbolic surfaces are only equivalent if there’s an isometry such that and .
So it turns out that for marked hyperbolic surfaces there isn’t this periodicity. When we go through a whole turn, the marking has changed by what’s called a Dehn twist, and we’ll talk about them in more detail later. But the point is that for marked hyperbolic surfaces, we can actually distinguish the gluings by more than just the angle, and this gives us some more coordinates.
Theorem: For each pair of pants decomposition of , each point of is associated to a marked hyperbolic surface with described lengths and twist parameters. Furthermore, every marked hyperbolic surface occurs this way.
These are called the Fenchel-Nielsen Coordinates and the space of marked hyperbolic surfaces is called Teichmüller Space. Topologically, this space, which we’ll denote , is very simple: it’s a product of copies of the real numbers with the positive real numbers, and so is contractible!
We’ll talk just a bit about the topology before calling it a day, though. We can actually define the topology on directly, and it turns out to be the same. Let be a marked hyperbolic surface and (just typing this feels like I’m breaking a self-imposed rule, but I swear, we’re not about to go into the jungle of estimates and asymptotics, though they certainly relate here! Mostly because I have other things to talk about) we can define as the set of surfaces such that for every simple closed curve $c\subset S$, we have . These manage to be a basis for the topology induced by the Fenchel-Nielsen coordinates.