BI: the Bestvina-Bromberg-Fujiwara construction

This post is about a remarkable construction due to Bestvina-Bromberg-Fujiwara. It has been first used to show that the asymptotic dimension of Mapping Class Groups is finite, and it is quite useful, for example, in this great paper by Dahmani-Guirardel-Osin (and I used it here, here and here, after all this is my blog 😉 ). “Call BBF! You give us projections, we give you a quasi-tree-graded space.”

The tl;dr description is: The input of the construction is a family of geodesic metric spaces and “projection maps” between them, and the output is a metric space that contains all the metric spaces you started with. Such space is hyperbolic if you started with (uniformly) hyperbolic spaces. More in general, it is relatively hyperbolic, and actually even quasi-isometric to a tree-graded space with pieces your original metric spaces (this is due to David Hume).

Let me start by describing a simple example that you might wish to keep in mind. Consider this curve on the surface $S$: Now consider all its lifts to $\mathbb H^2$, the universal cover of $S$. Schematically, the picture looks like this: Now, consider the closest-point projections onto the lines in picture. As it turns out, the projection of one line onto another one has uniformly bounded diameter. Also, if two lines project far away onto another line, then any geodesic connecting them behaves like the blue curve in this picture: That is to say, such geodesics pass close to the projection sets. What the BBF construction does in this case is roughly the following. You define a line $\gamma$ to be projection-between two other lines $\gamma_1,\gamma_2$ if the projections of $\gamma_1$ and $\gamma_2$ onto $\gamma$ are far away, as in the picture above. (This is highly *un*related to being “topologically” between.) Then, you form a path metric space in the following way: You consider the disjoint union of all lines, then you add paths of length 1 connecting all points in $\pi_{\gamma_1}(\gamma_2)$ to all points in $\pi_{\gamma_2}(\gamma_1)$ whenever there is no line projection-between $\gamma_1$ and $\gamma_2$. (This is not entirely correct, but close enough.) The construction is natural in the sense that the fundamental group of the surface we started with acts on the metric space we constructed. Also, as it turns out, such space is a quasi-tree, i.e. it is quasi-isometric to a tree.

As an aside, if we start from a loop in a (say, closed) hyperbolic 3-manifold $M$ instead, then we again end up with an action of the fundamental group of $M$ on a quasi-tree $T$. It is easy to see that there are elements of $\pi_1(M)$ acting loxodromically on $T$: The element of $\pi_1(M)$ corresponding to the original geodesic has an invariant line, i.e. the same invariant line it has in $\mathbb H^3$. This is quite interesting, because there are $M$‘s so that every action of $\pi_1(M)$ on a genuine tree fixes a point…

Ok, time to state the BBF axioms. I’ll mostly keep the same notation they use in their paper, which is motivated by curve complexes and subsurface projections. Let $\{\mathcal C(Y)\}_{Y\in \bf Y}$ be a collection of geodesic metric spaces and for each $Y$ let $\pi_Y$ be a map assigning to each $Z \in{\bf Y} \setminus \{Y\}$ a bounded subset of $Y$. The axioms are as follows (all axioms start with “There exists $\xi\geq 0$” and assume that all projections involved are well-defined).

1. The diameter of $\pi_Y(Z)$ is bounded by $\xi$ for each $Y\neq Z$.

Ok, that was easy.

Let’s set for convenience $d_Y(X,Z)=diam_Y(\pi_Y(X)\cup \pi_Y(Z))$ (i.e. “the distance between $X$ and $Z$ from the point of view of $Y$“), where $diam$ is the diameter.

2. Given $X,Z$, there exist only finitely many $Y$‘s so that $d_Y(X,Z)\geq \xi$.

Another way of saying this: There are finitely many $Y$‘s that are projection-between $X$ and $Z$. In the example of the lines in $\mathbb H^2$, this axiom is true because any geodesic from one line $\gamma_1$ to another line $\gamma_2$ fellow-travels for a bit all lines projection-between $\gamma_1$ and $\gamma_2$. The length of any such geodesic is finite, so… Ok, last axiom!

3. At most one of $d_Y(X,Z), d_X(Y,Z), d_Z(X,Y)$ can be larger than $\xi$.

A more compact way of formulating it is: $\min\{d_Y(X,Z), d_X(Y,Z)\}\leq \xi$. I formulated it in the other way because it makes it clearer what this axiom is useful for, i.e., ensuring that “being projection-between” behaves like the relation “being between” in a partial order. What I mean is that if $Y$ is between $X$ and $Z$ then, say, $X$ is not between $Y$ and $Z$.

Exercise: Show that the third axiom holds in the example, using the picture about far-away projections.

Finally, as mentioned earlier, the construction spits out a metric space $\mathcal X({\bf Y})$ constructed from the union of the $\mathcal C(Y)$‘s and some paths of length 1. The “right” statement about this metric space is that it is quasi-isometric to a tree-graded space whose pieces are copies of the $\mathcal C(Y)$‘s (proven by David Hume). In particular, if the $\mathcal C(Y)$‘s are (uniformly) hyperbolic, then $\mathcal X({\bf Y})$ is hyperbolic, and if the $\mathcal C(Y)$‘s are (uniformly) quasi-trees, then $\mathcal X({\bf Y})$ is a quasi-tree (proven by BBF). Also, $\mathcal X({\bf Y})$ is hyperbolic relative to the copies of the $\mathcal C(Y)$‘s that it contains. Just to wrap this all up, the BBF construction allows to construct interesting actions on hyperbolic and relatively hyperbolic spaces whenever projections with good properties can be defined. And if you’re not convinced that good actions on hyperbolic metric spaces are useful you can take a look at the paper by Dahmani-Guirardel-Osin, or wait for a future post about it. 😉