An even shorter proof that curve graphs are hyperbolic

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 $\delta\geq 0$ so that for any closed surface $S$ of genus at least 2 the curve graph $\mathcal{C} (S)$ of $S$ is $\delta$ -hyperbolic.

The proof relies on a “guessing geodesics lemma” similar, but simpler, to the one described here. It is Proposition 3.1 in this paper by Bowditch. (I plan to report on it in a later post.)

[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 $X$ be a metric graph and suppose that we assigned to each pair of points $x,y\in X$ a connected set $A(x,y).$ Suppose that there exists $D$ so that

1) if $x,y$ are at distance at most 1 then the diameter of $A(x,y)$ is at most $D,$ and

2) the $A(x,y)$ ‘s form thin triangles, i.e. $A(x,y)$ is contained in the $D$ -neighborhood of $A(x,z) \cup A(z,y)$ for each $x,y,z.$

Then $X$ is hyperbolic, with hyperbolicity constant depending on $D$ only.

For mere technical convenience, we will work with the “augmented curve graph” $\mathcal{C}_{aug}(S)$ where the edges connect curves that (in minimal position) intersect at most once. It is easy to see that $\mathcal{C}_{aug}(S)$ is quasi-isometric to $\mathcal{C}(S)$ (with constants not depending on $S$ ) because if two curves intersect at most once then their distance in $\mathcal{C}(S)$ is at most 2.

Bicorn curves. So, we need to guess, given $a,b,$ a family of curves forming a path from $a$ to $b.$ And the way to do this is “interbreeding” $a$ with $b$ : we put them in minimal position and define $A(a,b)$ to be the collection of all (simple closed) curves $c$ consisting of the union of an arc $a'$ of $a$ and an arc $b'$ of $b,$ which we call the $a$ -arc and $b$ -arc of $c.$ We also add $a,b$ to $A(a,b),$ and we call the elements of $A(a,b)$ bicorn curves.

Connectedness. First of all, we show that (the full subgraph of $\mathcal{C}_{aug}(S)$ spanned by) $A(a,b)$ is connected. There is a natural partial order on $A(a,b)$ : we write $c<c'$ if the $b$ -arc of $c'$ contains the $b$ -arc of $c,$ like in the following picture:

In other words we measure progress towards $b$ by looking at the how close the $b$ -arc is to being the whole of $b.$
So, the reason why $A(a,b)$ is connected is that for every $c\in A(a,b)$ there is $c'$ adjacent to $c$ and so that $c<c'$ (unless $c=b$ ). In particular, every point in $A(a,b)$ can be connected to $b.$
In order to see that such $c'$ exists start with $c$ and prolong one side of the $b$ -arc until it hits the $a$ -arc. The prolonged $b$ -arc is part of a bicorn curve $c'$ that (when put in minimal position) intersects $c$ at most once, and clearly $c< c'.$

As an aside, I like to think of $A(a,b),$ endowed with the partial order, as a “directed multi-path”.

Thinness. Property 1) from the proposition is straightforward: if $a,b$ are adjacent in $\mathcal{C}_{aug}(S)$ then $A(a,b)=\{a,b\}.$ Property 2) is also rather easy. Fix curves $a,b$ and a bicorn curve $c\in A(a,b).$ Given another curve $d,$ we have to find some $e\in A(a,d) \cup A(d,b)$ close to $c.$

The procedure is as follows. Put all curves in minimal position and consider three consecutive intersections of $d$ with $c$ (if there are fewer than three intersections then $d$ is close to $c$ and we are done). The picture might look like this, for example:

We can assume that two of those intersections $p,q$ are on, say, the $a$ -arc. The subarc of $d$ from $p$ to $q$ is part of a bicorn curve $e$ in $A(d,a)$ that (when put in minimal position) intersects $c$ at most twice. Therefore, $e$ and $c$ are within distance two of each other.