I love talks that start with “I haven’t seen this written down explicitly anywhere, but…” because that means someone is about to explain some math folklore! There are some statements floating around in mathland that specialists in those fields believe, so the rest of us believe them because the specialists said so, but no one knows a citation or a written proof for them. That’s folklore. Two weeks ago I gave a talk and someone asked a question (are RAAGS uniquely determined by their defining graphs?) and I said “probably, but I have no references for you.” I found a reference a day later and emailed it to her, but the reference was way hard and had way more machinery than I was expecting. The power of folklore!
Anyways, this series of posts will be based on a talk by a grad student at UT, Aaron Fenyes. This was a great, great talk with lots of pretty slides, which Aaron has generously allowed me to put up here. We’ll review curvature and surfaces, and then talk about how to go back and forth between two kinds of surfaces.
We’ve chatted about hyperbolic space v. Euclidean and spherical space before in terms of Euclid’s postulates, but let’s chat a bit about curvature. We say the Euclidean plane/real space has curvature 0, that hyperbolic space is negatively curved, and spherical space is positively curved. There’s a nice way to see curvature: draw a triangle in your space (that old link also has some triangle conditions in it), and imagine standing at a point on that triangle and looking toward one corner of the triangle. By “looking out” I mean your gaze should lie tangent to the triangle. Remember:
Now walk toward the corner you’re facing, and then walk down the second side of the triangle still facing that direction (so you’re sidestepping), and walk around the next corner (so you’re now walking backwards) and keep going until you end up where you started. This is called parallel transport. If your triangle was in Euclidean space, then you’re facing the same way you were when you started.
If your triangle was slim, then you might find yourself facing the opposite way that you started! Or if your triangle isn’t that curved, you’ll find yourself facing a direction counterclockwise from your original one.
Similarly, if your triangle was fat, you’ll end up facing a direction clockwise from your original.
Here’s the picture of just the first and last arrows:
So curvature is a way to measure how far clockwise you’ve turned after doing this parallel transport.
I love that description of curvature vs. the way I did it before, but they’re all good ways of seeing the same thing. Next we need to review surfaces. When we first met hyperbolic surfaces, we built them by gluing pairs of pants together, which themselves were stitched together from right angled hexagons which lived in hyperbolic space. Redux:
Now if I take a little patch from my hyperbolic surface, I can trace back through one or two pairs of pants to find the original hexagon(s) in hyperbolic space where my patch came from. So I have a map from hyperbolic space to my patch of hyperbolic surface, describing the metric and geometry around that patch. This map is called a chart, and every point on a hyperbolic surface will have a chart associated with it, sending some part of hyperbolic space to a neighborhood of that point.
This picture might make you leery: what happens when images of charts overlap, like they do here? The preimages in the hyperbolic plane are disjoint, but they map to the same yellow area in the surface. We want to say there’s some reasonable relationship between the yellow preimage patches in the hyperbolic plane. That relationship is the only one we know, isometry:
If we look only at the yellow patch, we can find another way to describe the map in the picture: first, do the blue chart sending the blue patch to the surface. Then, do the inverse of the orange chart, which sends the orange surface patch to its preimage. Restricted to the yellow overlap patch, this is the definition of a transition map.
So here’s another way to think of hyperbolic surfaces, instead of gluing hexagons together like before. 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.
If you change where the chart is coming from, we can change the adjective before surface. For instance, a flat surface is when the charts come from the Euclidean plane. Now we’re going to define half translation surfaces, where the charts come from the Euclidean plane, but we have some more conditions on the transition maps. The isometries of the Euclidean plane all come from a combination of translations and rotations. Instead of allowing all isometries, we’ll only allow some of them:
In this picture you can see the orange and blue patches on the surface which come from the Euclidean plane. Now we’re allowing translations and pi (180 degree) rotations only for our transition maps. That’s why they’re called half-translation surfaces: charts from the Euclidean plane, and transition maps are translations plus half-rotations (flips). As an aside, a translation surface is when we allow translations only, and no flips.
In the next post in this series, I’ll go through Aaron’s explanation of how we can go from hyperbolic surfaces to half-translation surfaces and back, and we’ll get to revisit our old friend the curve complex. It’ll be fun!