In this post I’d like to tell you about the definition of relative hyperbolicity I gave in this paper.
It is a definition that falls into the first family of definitions I described in this post, i.e. “relatively hyperbolic groups look like tree-graded spaces” (I promise that the next post will not involve tree-graded spaces, please keep reading my blog… please!).
The definition was inspired by two facts. First, closest point projections onto pieces are powerful technical tools to deal with tree-graded spaces, see section 2 here. Second, closest point projections onto horospheres are powerful technical tools to deal with horoball complements, one of the motivating examples of relative hyperbolicity.
Actually, such projections are heavily used in the proof by Farb that fundamental groups of (technical assumptions) finite volume negatively curved manifolds are relatively hyperbolic.
So, the idea is to characterize tree-graded spaces in terms of projections and then take a coarse version of this characterization.
So, let be a piece in a tree-graded space, and let be the closest point projection on . First property, by definition:
As it turns out, is locally constant in balls disjoint from . But I prefer to give you the following stronger property:
2) if then any geodesic from to contains and .
This property is the most important one, it happens to be crucial here as well (well, some version of it), in a quite different and general context. Finally:
3) if are distinct pieces then is a point.
One would like to say that a space is tree-graded with respect to a collection of subsets if and only if for each subset there exists satisfying the said properties. A piece is missing: no condition guarantees that the collection of subsets is, say, non-empty. So you have to add this boring condition called being transverse free which requires that geodesic triangles intersecting each piece in at most one point are tripods.
The projection properties plus this additional condition actually characterize tree-graded spaces.
Ok, we have our characterization of tree-graded spaces, and you can probably figure out the coarse versions of all projection properties. For example the coarse version of 2) is:
There exists so that whenever any geodesic from to intersects and .
And as it turns out the coarse versions of the projection properties plus the coarse version of the boring property characterize relatively hyperbolic groups. Actually, more in general, they characterize (metrically) relatively hyperbolic spaces, also called asymptotically tree-graded spaces. In such spaces one has a specified collection of subsets, called peripheral sets, which play the role of left coset of peripheral subgroups in relatively hyperbolic groups.
(The coarse version of the boring property is actually pretty annoying to state, the idea is that if a geodesic triangle intersects neighborhoods of the pieces in small sets then the triangle has small hyperbolicity constant.)
Ok, what are projections good for? Well, they allow you to control where geodesics go.
The basic example is the following. If you have satisfying the coarse versions of 1) and 2) then is quasi-convex (more than this, actually). I learned the idea for showing this and similar things from the paper by Farb on relative hyperbolicity. So, suppose that a geodesic with endpoints on goes far from . Then it has a long subgeodesic outside a neighborhood of of radius, say, . So, when you project subgeodesics of of length onto they get shrinked by a factor at least 10. It is not hard to believe that you get a contradiction with being a geodesic from this, if is long enough.
As it turns out, one can play around with several variations of this idea to get more sophisticated results. Especially if one exploits the nice interaction between projections and saturations. The saturation of a geodesic is the union of and all the left cosets of peripheral subgroups (peripheral subsets in the metric context) such that contains a long enough subgeodesic contained in a suitably chosen neighborhood of . Unfortuntely, one has to deal with 2 constants to define them formally, but I hope that the idea is clear: it is the coarse version of “the geodesic intersects the piece non-trivially” which works for tree-graded spaces. Here is the saturation of a geodesic in a tree-graded space, in red:
As it turns out there are projections on saturations as well, and if you really want to know it this is because adding to the collection of peripheral sets a saturation while removing all the pieces contained in it actually gives once again a relatively hyperbolic structure, in the metric sense. (This is why I brought up metric relatively hyperbolic structures.) Exercise: show this fact in tree-graded spaces.
So, using the quasi-convexity I told you about, one gets that saturations allow you to control geodesics. This turns out to be handy if you’re tampering with the metric and you want to keep some control of the geodesics, as in the applications I have in the paper. I must say that saturations are used in the paper by Cornelia Druţu and Mark Sapir that I cite compulsively.
Final remark. Hyperbolic elements in relatively hyperbolic groups are those infinite order elements that are not conjugate into a peripheral subgroup (in another language: they are not parabolic). As it turns out, their orbits are quasi-geodesics and there are projections satisfying the coarse versions of 1) and 2) onto them.
(The reason for this is that any hyperbolic element is contained in an elementary subgroup that can be added to the list of peripheral subgroups preserving relative hyperbolicity.)
Hyperbolic elements are quite interesting for several reasons and in several senses they are the analogues of pseudo-Anosovs in mapping class groups, but I’ll get back on this when I’ll tell you about this paper of mine and its bigger brother by Dahmani, Guirardel and Osin.