Last week we learned about covering spaces, and I made a promise about what we’d talk about in this post. For those who are more advanced, this all has to do with Scott’s separability criterion, so you can take a look back at that post for a schematic. I’ll put the picture in right here so this post isn’t all words:
Theorem (Patel): For any closed geodesic g on a compact hyperbolic surface of finite type with no cusps, there exists a cover such that g lifts to a simple closed geodesic, and the degree of this cover is less than , where is a constant depending on the hyperbolic structure .
We know what geodesics are, and we say they’re closed if the beginning and end are the same point (so it’s some sort of loop, which might intersect itself a bunch). But wait, Yen, I thought that geodesics were the shortest line between two points! The shortest path from a point to itself is not leaving that point, so how could you have a closed geodesic? Nice catch, rhetorical device! A closed geodesic is still going to be a loop, but it won’t be the shortest path between endpoints because there are no endpoints. Instead, just think locally: if a closed geodesic has length l, then if you look at any two points x and y less than l/2 apart from each other, the closed geodesic will describe an actual geodesic segment between x and y. It’s locally geodesic.
What about hyperbolic surfaces of finite type with no cusps? Well, we say a surface is of type (g, b, n) if it has genus g (that’s the number of holes like a donut), b boundary components, and n punctures or cusps.
Boundary components are sort of like the horizontal x-axis for the half plane: you’re living your life, totally happy up in your two-dimensional looking space, and then suddenly it stops. This is also what a boundary of a manifold is: where the manifold locally looks like a half-space instead of all of . Surfaces are 2-manifolds.
Finally, I drew punctures or cusps suggestively- these are points where you head toward them but you never get there, no matter how long you walk. These points are infinitely far from the rest of the surface.
I think we know all the rest of the word’s in Priyam’s theorem *(hyperbolic structure is a hyperbolic metric). The important thing to take from it is that she bounds the degree of the cover above by a constant times the length of the curve. This means that she finds a cover with degree smaller than her bound (you can always take covers with higher degree in which the curve still embeds, but the one she builds has this bound on it).
She’s looking for a minimum degree cover and finds an upper bound for it in terms of length of the curve. Let’s write that as a function, and say gives you the minimum degree of a cover in which curves of length L embed (using the hyperbolic structure ). What about a lower bound?
Theorem (Gupta, Kapovitch): If is a connected surface of genus at least 2, there exists a such that for every .
So they came up with a lower bound, which uses a constant that depends on both the surface and the structure. But it looks like it only works on curves that are long enough (longer than the systole length, which we’ve seen before in Fanoni and Parlier’s research: the length of the shortest closed geodesic on the surface). Aha! If you’re a closed geodesic, you’d better be longer than or equal to the shortest closed geodesic. So there isn’t really a restriction in this theorem. Also, that paper is almost exactly 1 year old (put up on arxiv on 11/20/2014).
Now we have .
This is where it gets exciting. We know from Scott in 1978 that this all can be done, and then Patel kickstarts the conversation in 2012 about quantification, and then two years later Gupta and Kapovich do the other bound, and boom! in January 2015, just three months after Gupta-Kapovich is uploaded to the internet, my buddy Jonah Gaster improves their bound to get , where his constant c doesn’t even depend on . He does this in a very short paper, where he uses specific curves that are super hard to lift and says hey, you need at least this much space for them to not run into each other in the cover.
Here’s a schematic of the curves that are hard to lift (which another mathematician used to prove another thing [this whole post should show you that the mathematical community is tight]):
So that’s the story… for now! From Liverpool (Peter Scott) to Rutgers in New Jersey (Priyam) to Urbana/Champaign in Illinois (Gupta and Kapovitch) to Boston (Jonah), with some quick nods to a ton of other places (see all of their references in their papers). And the story keeps going. For instance, if you have a lower bound in terms of length of a curve, you automatically get a lower bound in terms of the number of times it intersects itself (, same mathematician who came up with the curves). So an open question is: can you get an upper bound in terms of self-intersection number, not length?