In this post I’ll talk about this paper. There are many groups that have been known for quite a while to “look like” hyperbolic groups, but only in certain “directions”. Examples of such directions are (the axes of ) rank 1 isometries in CAT(0) groups, pseudo-Anosovs in mapping class groups (of closed surfaces), elements acting hyperbolically in groups acting acylindrically on a simplicial tree, iwip (aka fully irreducible) elements of .
I left out the example I mostly had in mind when writing the paper: hyperbolic elements (of infinite order) in relatively hyperbolic groups. Those are the infinite order elements of a relatively hyperbolic group not conjugated into any peripheral subgroup.
They generate undistorted subgroups that have little in common with the peripheral subgroups, something like this:
The picture represents a tree-graded space approximating a portion of a relatively hyperbolic group, with the red line representing the orbit of the hyperbolic element while the flats represent left cosets of peripheral subgroups.
Hyperbolic elements, as well as all the other examples mentioned above, are known to be Morse, which means the following. First, their orbits are quasi-geodesics, and second whenever you have a blue quasi-geodesic joining points on a red orbit, the quasi-geodesic stays close to the orbit:
Indeed, a stronger property holds. As it turns out, a closest point projection on the orbit of a hyperbolic element has the property that
(*) if is large enough then all geodesics from to pass close to and .
Exercise: this is indeed a stronger property. Hint: show that a path connecting to and staying far from the orbit is very long compared to a geodesic connecting to .
So, the natural question is: do we have this property, say, for mapping class groups and pseudo-Anosovs or fundamental groups of graph manifolds and elements acting hyperbolically on its Bass-Serre tree, given a word metric? I don’t know… I wildly guess that it’s not true in general, but nonetheless a version of (*) holds. The idea is that there’s nothing special about a given word metric, so there’s nothing special about geodesics in a given word metric. We might as well consider a family of quasi-geodesics (with uniformly bounded constants) so that every pair of points can be joined by a quasi-geodesic in the family. The advantage of doing this is that in certain cases you have a family of special quasi-geodesics with nice description and properties that are easier to handle. In the mapping class group case one can use hierarchy paths, while in the graph manifold case one can use the paths described here. (Probably in the mapping class group case one can also use splitting sequences in the train tracks complex, as defined in this paper, but I’m not very familiar with them…)
The map is no longer defined as the closest point projection, one just says that the set (for example, the orbit of an element) is contracting if there exists a map restricting (coarsely) to the identity on so that (*) holds when the word “geodesics” is replaced by “special quasi-geodesics”.
This turns out to work in most of the cases mentioned at the very beginning of the post, meaning that, in all those examples of pairs group/family of elements, the elements in the family have quasi-geodesic and contracting orbits ( and groups acting acylindrically need to be treated in a slightly different way).
Here is what I used this for:
Thereom: If a finitely generated subgroup contains a contracting element and is not virtually cyclic, then a (simple) random walk on the subgroup ends up in a non-contracting element with probability decaying exponentially in the length of the random walk.
In other terms, if you write down a long random word in the generators of the subgroup, the probability that this word represents a non-contracting element is small.
Special cases of this theorem were known already, most notably it was known for mapping class groups.
In order to show the theorem one has to use just one contracting element to produce many others. The lemma that allows to do this states that, loosely, if is a contracting element and is “generic” then is contracting for a suitable . By “generic” I mean the following. It turns out that every contracting element is contained in a maximal virtually cyclic subgroup so that whenever is not in then the projection of on has uniformly bounded diameter . Generic means not in .
So, in the setting of the lemma, we have the following picture:
All the drawn translates of an orbit of have very little in common, by genericity, so it should not be a surprise that the red line is a quasi-geodesic. The map that witnesses the fact that is contracting can be defined looking at the projections of a given on the translates of .
Given this way of constructing contracting elements, it’s not very hard to prove the theorem. In a sufficiently long random word in the generating set for the subgroup you expect to see any given word, and in particular a word representing a contracting element . You might as well assume that this appears as a final subword, up to cycli conjugation. A subgroup as in the theorem turns out to contain many free groups, so there’s “a lot of space” and no reason at all for a random walk to stay close to a given virtually cyclic sub-subgroup. A similar argument applied to the random sub-walk before the final subword gives that we are in the setting of the lemma described above.
One has to be a little bit more careful with the estimates (and possibly repeat this argument) but there really is not much more than this in the formal proof.