In this post I described the Hensel-Przytycki-Webb proof that curve graphs are (uniformly) hyperbolic. Their methods actually apply to the arc graph, and then they used a clever evil trick due to Harer that involves adding an artificial puncture to go from the hyperbolicity of arc graphs to that of curve graphs.
After discussing with Piotr Przytycki, we came up with a proof that takes place directly in the curve graph. To be more precise, the proof works in the closed case, i.e. exactly when the HPW argument requires the additional trick.
Here is the statement that we are going to prove.
Theorem: There exists so that for any closed surface of genus at least 2 the curve graph of is -hyperbolic.
[EDIT: it has been pointed out to me that the criterion was actually discovered by Masur-Schleimer, see Theorem 3.15 here.]
Informally speaking, it says that if a space contains a family of paths forming thin triangles, then it is hyperbolic.
Proposition (Masur-Schleimer): Let be a metric graph and suppose that we assigned to each pair of points a connected set Suppose that there exists so that
1) if are at distance at most 1 then the diameter of is at most and
2) the ‘s form thin triangles, i.e. is contained in the -neighborhood of for each
Then is hyperbolic, with hyperbolicity constant depending on only.
For mere technical convenience, we will work with the “augmented curve graph” where the edges connect curves that (in minimal position) intersect at most once. It is easy to see that is quasi-isometric to (with constants not depending on ) because if two curves intersect at most once then their distance in is at most 2.
Bicorn curves. So, we need to guess, given a family of curves forming a path from to And the way to do this is “interbreeding” with : we put them in minimal position and define to be the collection of all (simple closed) curves consisting of the union of an arc of and an arc of which we call the -arc and -arc of We also add to and we call the elements of bicorn curves.
Connectedness. First of all, we show that (the full subgraph of spanned by) is connected. There is a natural partial order on : we write if the -arc of contains the -arc of like in the following picture:
In other words we measure progress towards by looking at the how close the -arc is to being the whole of
So, the reason why is connected is that for every there is adjacent to and so that (unless ). In particular, every point in can be connected to
In order to see that such exists start with and prolong one side of the -arc until it hits the -arc. The prolonged -arc is part of a bicorn curve that (when put in minimal position) intersects at most once, and clearly
As an aside, I like to think of endowed with the partial order, as a “directed multi-path”.
Thinness. Property 1) from the proposition is straightforward: if are adjacent in then Property 2) is also rather easy. Fix curves and a bicorn curve Given another curve we have to find some close to
The procedure is as follows. Put all curves in minimal position and consider three consecutive intersections of with (if there are fewer than three intersections then is close to and we are done). The picture might look like this, for example:
We can assume that two of those intersections are on, say, the -arc. The subarc of from to is part of a bicorn curve in that (when put in minimal position) intersects at most twice. Therefore, and are within distance two of each other.
And this is it. Yep, really.
We welcome suggestions about further applications of bicorn curves…