Last week we saw the standard definition for a hyperbolic surface. You can tweak this standard definition to define all sorts of surfaces, and we tweaked it for a definition of half-translation surfaces. Here are the two definitions:
- A hyperbolic surface is a topological space such that every point has a neighborhood chart from the hyperbolic plane and such that the transition maps are isometries.
- A half-translation surface is a topological space such that all but finitely many points* have a neighborhood chart from the Euclidean plane and such that the transition maps are combinations of translations and flips. These finitely many points are called singularities.
Precision note: according to Wikipedia, we need to add the adjective “Hausdorff” to our topological space. We won’t worry about this or give a precise definition of it; you can just know that Hausdorff has something to do with separating points in our space.
Half-translation spaces come with something nifty that occurs in Euclidean space. You know how when you look at a piece of notebook paper, it has all these nice parallel lines on it for writing? Or if you look at a big stack of paper, each sheet makes a line which is parallel to the hundreds of others?
Mathematicians call that a foliation: each sheet of paper is called a leaf. This is an intuitive definition; we’re not going to go into a technical definition for foliation. Just know that Euclidean space comes with a foliation of all horizontal lines y=r, where r is some real number. Then since transition maps of half-translation spaces come from either straight translations or flips, the foliation carries over to the half-translation space (though orientation might have flipped, we don’t care about those in this application).
Notice in the picture in the lower left that there are a few points where the horizontal foliation doesn’t quite work. Those are the singularities that show up in the definition of a half-translation surface (we need them if we want our surface to be anything besides an annulus).
At those singular points, we glue together patches of Euclidean space. The orange color in this picture shows the path of the critical leaf as it winds all the way around the surface some number of times.
Last week we had those nify gifs to show us how to think about curvature as positive, negative, or zero. Here’s the example of zero curvature, because the last arrow is the same as the first arrow:
We can actually get precise numbers instead of just signs for curvature.
Here the triangle encompasses -π/3 curvature. Notice that it embeds straight down into the hyperbolic surface, so we see an actual triangle down in the lower left. If we made this triangle bigger and bigger, eventually it’d wrap around the surface and we wouldn’t see a triangle, just a bunch of lines hinting at a triangle up in the hyperbolic plane. That’s the next picture.
Curvature can range from -π to π. Here’s an example of an extremely negatively curved triangle which has -π curvature:
Such a triangle in hyperbolic space has all three corners on the boundary/at infinity. This is called an ideal triangle. So all ideal triangles encompass -π curvature. You can see also how in the surface, we have a collection of lines whose preimage is the ideal triangle.
We can also use the same process to find curvature in other places. For instance, if we make a little hexagon around a singularity of a half-translation surface, we can go around it with the same parallel transport process to figure out how much curvature the singularity contains. We’ll make use of that horizontal foliation we saw earlier.
This looks very similar to our ideal triangle: the arrow starts off pointing up, and ends up pointing exactly the opposite direction. So this singularity has -π curvature too, just like the ideal triangles.
Now for the math part! Here’s the question: given a hyperbolic surface, how can we construct an associated half-translation surface?
Answer: we’ll use those foliations that we had before, as well as something called a geodesic lamination: this is when you take a closed subset of your surface, and give it a foliation. So it’s like a foliation, only there’ll be holes in your surface where you didn’t define how the pages stack. The first example of a geodesic lamination is a plain ol’ geodesic curve in your surface: the curve itself is a closed subset, and the foliation has exactly one leaf, the curve itself. After this example they get real funky.
Given a book, we might want to know how many pages we’ve read once we stick our finger in somewhere. Luckily there are page numbers, so we can subtract the page number we started at from the page number we’re standing at. Similarly, given a foliation, we might want to have a measure on it, transverse to the leaves. If we have one, it’s called a measured foliation. These exist.
So let’s start with our hyperbolic surface, and choose a maximal measured geodesic lamination on it. Maximal means that the holes are the smallest they could possibly be. Turns out this means they’re the images of ideal triangles under the atlas.
Also, there are only finitely many of these triangle-shaped holes down in the surface (we’re sweeping some math under the rug here). Now we need to get from this surface to a half-translation surface. We’ll keep that foliation given by the lamination, and we need to get rid of those complementary triangles somehow. So the lamination’s foliation will become the horizontal foliation of the half-translation surface, and the ideal triangles will correspond to singular points. We can’t just collapse the ideal triangles to singular points, because as we saw earlier, images of ideal triangles are really funky and wrap around the surface. We need to find smaller triangles to turn into singular points. Here’s the picture:
Upstairs, we made a new purple foliation (transverse to the lamination) of the complementary ideal triangles, by using arcs of circles perpendicular to the boundary circle (these circles are called horocycles). So now we have teensier triangles in the middle of the ideal triangles, called orthic triangles. To make a half-translation surface, we’ll quotient out the horocycles, which means that in each ideal triangle, we identify an entire purple arc with one point.
In this way we get tripods from triangles. The middles of these tripods are singular points of the half-translation surface. The measure from the measured lamination gives a measure on the foliation of the half-translation surface.
But Euclidean space actually comes with horizontal and vertical distances defined (remember, half-translation surfaces locally look like Euclidean space). So far we have a way to get one direction. How do we get the transverse distance? We use the fact that we chose a geodesic lamination of our hyperbolic surface. Geodesics are curves of shortest length; in particular they have length. So if I’m in my translation surface and moving along a leaf of the foliation, I can look back at where I was in the lamination of the hyperbolic surface and use that distance. [There’s some rug math here too.] So we’ve made neighborhoods in the half-translation surface look like Euclidean space.
So that’s that! You can also go backwards from a half-translation surface to a hyperbolic surface by blowing up the singular points into ideal triangles. [More math, especially when the singularities of the half-translation surface are messy or share critical leaves]. Aaron claims this is folklore, but a quick google search led me to this paper (in French) and this one by the same author who connects flat laminations (on half-translation surfaces) to the geodesic ones we see in hyperbolic surfaces in section 5.
*I lied about finitely many points. You can have infinitely many singularities in a half-translation surface; they just have to be discrete (so you should be able to make a ball around each other disjoint from the others, even if the balls are different sizes). Examples of discrete sets: integers, . Examples of not-discrete sets: rational numbers, .