This post is based on this paper with David Hume.
As discussed here, it is a nice property for a group to be quasi-isometrically embeddable in a product of finitely many trees. Graph manifolds are certain 3-manifolds, and if you don’t know what they are just bear in mind the following example: Take two surfaces with boundary, take the product of each with and glue the resulting manifolds along the toric boundary switching the factors.
These manifolds are interesting, to me at least, because if you want to prove a property of all 3-manifold groups invoking geometrization, graph manifolds are one of the cases you have to consider. They are somewhat less studied than the other cases so you can have a lot of fun with them.
Anyway, we showed:
Theorem: If is a graph manifold, then there exists a quasi-isometric embedding of into a product of trees.
3 is optimal for closed graph manifolds, and at first we suspected that it was the case for non-closed ones as well. But this is probably not the case, see below.
So, let’s construct a couple of trees given a graph-manifold , for simplicity say the one depicted above. Start off with a bi-coloring of the Bass-Serre tree of .
What you see above each vertex is a product of the universal cover of a compact surface with boundary and . The universal cover of the surface is a fattened tree, i.e. it looks like this:
Ok, so above each vertex we have, essentially, a product of a tree and . Now, we want to put together all the trees sitting above the red vertices of the Bass-Serre tree.
Take red vertices at distance 2. If you think about it, you see that the corresponding trees contain geodesics that can naturally be regarded as parallel. Let me try and help you to visualize this. Consider the “intermediate” vertex. Above it you can see a strip, i.e. an interval times , connecting the boundary components corresponding to the red vertices we are aconsidering. The boundary components of such strip naturally correspond to geodesics in .
So, you can construct a tree starting with a disjoint union of all trees lying above red vertices, and identifying corresponding geodesics. And of course, you can also construct similarly .
There is a natural coarsely lipshictz map . The reason why this map might fail to be a quasi-isometric embedding is because it might fail to “see” some long geodesics that spend very little time above each vertex.
However, one can keep track of such geodesics in the Bass-Serre tree, and this actually is the third tree involved in the product.
I suspect that, at least for certain non-closed graph-manifolds, there is no need to “stabilize” taking the product with the Bass-Serre tree. One needs to arrange things in such a way that whenever a geodesic passes above a vertex, this gets recorded in either or , or a similar weaker property.