EO: quasi-isometric embeddings of the hyperbolic plane in relatively hyperbolic groups

This post is based on a joint work with John MacKay.

It is still an open question whether or not all non-virtually-free hyperbolic groups contain a surface group. This is actually related to (and AFAIK inspired by) an important open problem in hyperbolic geometry, the virtually Haken conjecture: is it true that all closed hyperbolic 3-manifolds have a finite-sheeted cover that contains an embedded surface? If this turns out to be true (I don’t know of anyone who’s convinced that it is not) then work of Dani Wise (in advanced preparation) would imply the virtual fibering conjecture: all closed hyperbolic 3-manifolds have a finite sheeted cover which fibers over $S^1$. And this is pretty awesome IMHO.
Well, I’m digressing, sorry about that, but I really like 3-manifolds.

Anyway, Bonk and Kleiner have a paper where they prove the “geometric analogue” of the open question I mentioned at the beginning:

Thereom: All non-virtually-free hyperbolic groups contain a quasi-isometrically embedded copy of $\mathbb{H}^2$.

The scheme of the proof is as follows. First, using some deep results on splittings of hyperbolic groups you reduce to the case of a one-ended hyperbolic group $G$.
Now you notice that if you have a quasi-isometric embedding of a “quadrant” in $\mathbb{H}^2$, that is to say the subspace pictured below in the disc model, into $G$ then you have quasi-isometric embeddings of balls in $\mathbb{H}^2$ of arbitrarily large radius. By an Arzela-Ascoli-type argument you see that this is enough to have a quasi-isometric embedding of $\mathbb{H}^2$. Now, the key point is the following: if you have an arc in the boundary of $G$ which is linearly connected (see below), you can use it to construct a quasi-isometric embedding of a quadrant in $\mathbb{H}^2$ into $G$. After all this, the problem is reduced to finding an arc with the following geometric property in $\partial G$:

A metric space $X$ (e.g. an arc) is said to be linearly connected if there exists $L$ so that for each pair of points $x,y$ in $X$ there is an arc of diameter at most $L d(x,y)$ connecting them. Notice that this is a quantitative version of being locally path-connected.
It not that hard to show that $\partial G$ is linearly connected, using the group action and the deep result that if $\partial G$ is connected then it is locally path-connected. Also, $\partial G$ is doubling, meaning that there exists $N$ so that each ball of radius $r$ can be covered by at most $N$ balls of radius $r/2$.
We are done, because of the following:

Theorem: any linearly connected, doubling (and complete) metric space contains a non-trivial linearly connected arc.

The theorem follows from a result of Assouad combined with a result of Tukia, but actually John found a much simpler proof.
Ok, it’s time to state our result. Call a relatively hyperbolic group silly if it splits as the fundamental group of a graph of groups where edge groups are finite and vertex groups are either finite or parabolic (e.g.: $\mathbb{Z}^2*\mathbb{Z}/5$).

Theorem: Suppose that $G$ is hyperbolic relative to virtually nilpotent subgroups. Then $G$ contains a quasi-isometrically embedded copy of $\mathbb{H}^2$ if and only if it’s not silly.

Idea of the proof: plagiarize what has been done for hyperbolic groups. You might wish to take a look at this post if you’re not familiar with the idea that any relatively hyperbolic group $G$ acts properly and with some other properties on a certain hyperbolic space $X(G)$.
We needed the hypothesis that the peripheral subgroups are virtually nilpotent because this guarantees that $\partial G=\partial X(G)$ is doubling (this is essentially due to Dahmani and Yaman). More on this later…
Also, boundaries of relatively hyperbolic groups with finitely presented one-ended peripherals are linearly connected. So, the exact same proof as above gives you an embedding of $\mathbb{H}^2$ into $X(G)$. This is not good enough, but it’s pretty close. We need the following new ingredient: if the image of the embedding avoids the horoballs (more precisely: its intersections with the horoballs have uniformly bounded diameter) then one can actually pass from the embedding in $X(G)$ to an embedding in $G$. So, we would like to translate this condition on avoiding horoballs into a condition on the boundary. Here’s the picture that allows us to do it: (The picture is a tree approximating an analogous configuration in $X(G)$.) Points in the horoball $O$ are in red. As you can see, a ray from a basepoint $w$ to $b\in\partial G$ “does not interfere much” with the horoball $O$ if $(a_O|b)\leq d(w,O)+const.$, where $a_O$ is the parabolic point in $\partial G$ corresponding to $O$ and $(\cdot|\cdot)$ is the Gromov product.
As the metric on the boundary is up to multiplicative error $e^{-\epsilon(\cdot|\cdot)}$, this translates into $d(b,a_O)\geq e^{-\epsilon d(w,O)}/const.$. So we want all points $b$ in the arc we are trying to construct to satisfy this property, for all $O$ and some constant.
In the paper we could not quote the result on existence of quasi-arcs directly, due to this extra condition we want, but indeed we could use essentially the same proof. I’m trying to convince John to write a post about his proof, I’ll keep you posted (lame pun intended).
Anyway the idea is as follows: you start with an arc, you modify it improving its geometric properties at a certain scale, repeat at smaller and smaller scales (a geometric sequence of scales) and take the limit. So, in order to obtain the required extra property, it’s enough at each scale to “detour” the arc around the parabolic points $a_O$ so that $e^{-\epsilon d(w,O)}$ is approximately the scale we are working at.
This detour operation is made possible by the very nice fact that the geometry of the peripheral subgroups control the local geometry of the boundary around parabolic points. Maybe I’ll give more details in another post, this one is getting long…
Anyway, a statement like:

Lemma: Let $H$ be finitely presented. Then for each $x,y\in H$ far enough from the identity there exists a path connecting them contained in $B_{3R}(1)\backslash B_{r/3}(1)$, where $R=\max\{d(x,1),d(y,1)\}$ and $r=\min\{d(x,1),d(y,1)\}$.

can be turned into a statement on “avoidability” of parabolic points described the following picture, where the central point is $a_O$ and $r$ is small compared to $e^{-\epsilon d(w,O)}$: Final note: the hypothesis that the peripherals are virtually nilpotent is very annoying. It seems too strong. In fact, it’s only used to get doubling boundaries, but boundaries are always “doubling far from parabolic points”, and you only work far from parabolic points. However, we were not able to implement this idea…