BI: Tree-graded spaces

Tree-graded spaces have been defined by Cornelia Druţu and Mark Sapir [DS]. They provide models for the coarse geometry of relatively hyperbolic groups, see example 2) below, but I’ll get back on this in some other post.
I’ll assume that the reader is familiar with (real) trees. If not, here’s a picture:

Denis Osin describes tree-graded spaces in the following way. A tree is a… tree, right? Well, a tree-graded space is a tree with apples:

Actually, I prefer to draw flat apples (possibly of infinite diameter):

Ok, what does this actually mean? A tree-graded space is a geodesic metric space with a specified family of closed geodesic subsets (the “apples”), called pieces. Distinct pieces intersect in at most one point, and each simple loop is contained in a piece.
As it turns out (formal statements are in Section 2, [DS]) tree-graded spaces are kind of built up in the following way (kind of because the situation is more complicated than what I’m about to describe, but what follows is definitely useful for the intuition). Start with a piece:

Then you attach trees to some of the points of the piece:

Such trees are called transversal trees. In order to get from one tree to another one you have to get back to the piece. Now attach pieces to some points on the transversal trees or the piece. Once again, the paths in the tree-graded space are the ones “you see in the picture”. Keep going.
Here is an example of a geodesic in a tree-graded space, in red:

Ok, examples of tree-graded spaces:
0) every space is tree-graded with respect to itself, this is sometimes called trivial tree-graded structure. The examples below are actually examples of non-trivial tree-graded structures.
0′) trees are tree-graded with respect to the empty set.

1) Cayley graphs of groups of the form $A*B$ with respect to the union of generating sets for the non-trivial groups $A$ and $B$ . These are tree-graded with respect to the left cosets of $A$ and $B$ .

2) asymptotic cones of relatively hyperbolic groups (this is why tree-graded spaces were defined in the first place), if the peripheral subgroups are proper subgroups. The pieces are copies of asymptotic cones of the peripheral subgroups, indeed they are ultralimits of left cosets of the peripherals. I’ll get back on asymptotic cones in another post.

3) recall that a cut-point $p\in X$ , where $X$ is, say, a geodesic metric space, is a point such that $X\backslash \{p\}$ has more than one path-connected component. If the geodesic metric space $X$ has a cut point, then the collection of all maximal connected subspaces without cut-points of $X$ is a tree-graded structure. This gives us the examples below.

4) asymptotic cones of mapping class groups (of, say, closed surfaces of genus $g\geq 1$ ), many fundamental groups of graphs of groups, right-angled Artin groups that are not products, $Out(F_n)$ ( $n\geq 2$ ),… However, the tree-graded structure is more complicated than in 2).

Finally, just an example of how tree-graded spaces can be used.

Lemma [DS]: if $A$ doesn’t have cut points and $X$ is tree-graded then any injective continuous map $f:A\to X$ has the property that $f(A)$ is contained in a piece.

The proof uses the fact that all paths connecting distinct pieces pass through a specified point in the first piece (and another specified point in the second piece, of course). So, $f(A)$ cannot contain, say, points contained in disjoint pieces.
One can use this lemma to show that automorphisms of a group hyperbolic relative to, say, copies of $\mathbb{Z}^2$ preserve peripheral subgroups up to conjugacy (if you’re not familiar with relative hyperbolicity just think of $\mathbb{Z}^2 * \mathbb{Z}^2$ , whose peripheral subgroups are the free factors). In order to prove this, recall that an automorphism induces a quasi-isometry of the Cayley graph, and taking asymptotic cones one turns this quasi-isometry into a bilipschitz homeomorphism. Using the lemma, one sees that such homeomorphism preserves the pieces, and it turns out that this can be used to show that peripheral subgroups are preserved up to conjugacy. For the details, see once again [DS].

More on tree-graded spaces is coming soon…

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