BI(+EO): quasi-isometric rigidity

There are a few types of quasi-isometric rigidity results. I will talk about results of the following kinds:
1) self-quasi-isometries of such-and-such space or group (coarsely) preserve some structure, usually a collection of (left cosets) of subgroups
2) if a group is quasi-isometric to such-and-such group/space then it is very similar to such-and-such group/space.

Examples:
1) (Druţu-Sapir) If $G$ is hyperbolic relative to subgroups isomorphic to $\mathbb{Z}^n$ , $n\geq 2$ , then every self-quasi-isometry (resp., every automorphism) of $G$ coarsely preserves left cosets of peripheral subgroups (resp. the collection of the conjugacy classes of the peripheral subgroups).
2) (Schwartz) if a group $G$ is quasi-isometric to the fundamental group of a finite volume hyperbolic manifold then there is a short exact sequence $1\to F\to G\to \pi_1(O)\to 1$ , where $F$ is finite and $O$ is a hyperbolic orbifold.

Those are not all possible types of rigidity results. For example, there are results of the type “all self-quasi-isometries of such-and-such space are at finite distance from a map of this form”, which are often intermediate between 1 and 2. Results of this form have been proven for certain symmetric spaces of rank one, where quasi-isometries are at finite distance from isometries, and for mapping class groups (of all but a few surfaces), where quasi-isometries are at finite distance from left multiplications.

Let us start with results of type 1, as they are very often preliminary to results of type 2. If you don’t know what asymptotic cones are, just think of them as machines to turn quasi-isometries into (bilipschitz) homeomorphisms. In particular, if you have some information about the topology of the asymptotic cones of some space you might deduce something about the self-homeomorphisms of the asymptotic cones, and this translates into information about the self-quasi-isometries of your space. Let us take the setting of example 1. Asymptotic cones of $G$ are tree-graded, and the pieces are homeomorphic to $\mathbb{R}^n$ . What you would like to show is that if you have a topological embedding of $\mathbb{R}^n$ into an asymptotic cone of $G$ then the image is a piece. As a consequence of this, self-homeomorphisms of the cone preserve the pieces, and the required result follows.

The proof is not difficult, and it is based on the picture below. If you have distinct pieces, there is a red point on one of them (well, both) so that every path connecting the pieces contains the red point.

Now, you see that the image of a topological embedding of $\mathbb{R}^n$ into the cone cannot contain, for example, the blue points. In fact, there are no cut points in $\mathbb{R}^n$ , i.e. no point disconnects $\mathbb{R}^n$ . (This is a lucky case, quite often it’s not enough to just consider topological embeddings of what you would like to be preserved.) Notice that the result in example 1 extends to groups hyperbolic relative to subgroups whose asymptotic cones don’t have cut points. These include non-virtually cyclic solvable groups, for example.

In this paper we (=Roberto Frigerio, Jean-François Lafont and myself) dealt with a somewhat similar setting. The manifolds we consider are built up from blocks consisting of the product of a finite volume hyperbolic manifold (with truncated cusps) of dimension at least 3 and a torus. So, in the universal cover of such a manifold of dimension $n$ we have copies of $\mathbb{R}^{n-1}$ , called walls, that cover the boundary tori of the blocks. The walls (turn out to be undistorted and hence) give rise to copies of $\mathbb{R}^{n-1}$ in the asymptotic cones, the $\omega-$ walls. Given a pair of $\omega-$ walls, instead of a red point we have a red subset, which has codimension at least 2.

So, once again, the image of a topological embedding of $\mathbb{R}^{n-1}$ in the cone cannot contain the blue dots, as otherwise we would have a subset of $\mathbb{R}^{n-1}$ which disconnects it and is homeomorphic to a closed subset of $\mathbb{R}^{n-3}$ . It is intuitively clear that such a thing does not exist, and a proof can be found for example in this lovely short note.

Ok, time to discuss type 2. Suppose that you have a group $G$ quasi-isometric to some space $X$ . In $G$ you have a natural family of isometries, given by left multiplications. Using the given quasi-isometry these turn into self-quasi-isometries of $X$ . This gives you an “approximate action” of $G$ onto $X$ by quasi-isometries with uniform constants, and you would like to extract information about $G$ out of this. In example 2, $X$ is the universal cover of a hyperbolic manifold of finite volume with truncated cusps, also called neutered space. As it turns out, quasi-isometries of such an object are at finite distance from the restriction of isometries of $\mathbb{H}^n$ (the first step in the proof of this being that covers of the boundary tori are coarsely preserved, that it to say example 1). So, you can promote the quasi-action on $X$ to an honest action by isometries on $\mathbb{H}^n$ , and then you’re in good shape.

In the case of the manifolds we considered (which we called higher-dimensional graph manifolds, by the way), we can show that a self-quasi-isometry of the universal cover coarsely preserves the coverings of the blocks. This follows quite easily from the fact that walls are coarsely preserved, as we kind of discussed earlier. So, this allows you to reduce to studying groups $H$ quasi-isometric to the universal covers of the blocks. Those are of the form (neutered space) $\times \mathbb{R}^k$ and as it turns out you can “project” the action on the neutered space factor. So, you are now in a setting very similar to that of example 2 and you can deduce that there exists a short exact sequence of the same form for the quotient of $H$ by the kernel of the projection. Finally, you show that the said kernel is quasi-isometric to some $\mathbb{R}^{k}$ , which by a result of Gromov implies that it is virtually $\mathbb{Z}^{k}$ . At the end of the day you get a result of the form: if a group is quasi-isometric to the fundamental group of a higher dimensional graph manifold then it splits as a graph of groups where vertex groups virtually satisfy $1\to \mathbb{Z}^{k}\to G_v\to \Delta\to 1$ , where $\Delta$ is as in example 2.