Here’s a recent preprint that sounds pretty interesting by Behrstock, Drutu and Sapir . The asymptotic cone of a metric space X is a new metric space that one constructs by scaling the metric on X by smaller and smaller numbers (i.e. you define for small s) and taking a limit as the scaling factor goes to zero. (Actually, you take an ultralimit, which is determined by an ultrafilter, which I won’t explain here. But I do plan on trying to use the prefix “ultra” in my own definitions whenever I can.) The asymptotic cone is a popular construction in coarse geometry because when you shrink your metric like this, the coarse features of the space turn into Lipschitz features. Whatever is left in the limit is completely determined by the large scale geomety. For example, the asymptotic cone of a delta-hyperbolic space is always a tree. The asymptotic cone of Euclidean space is Euclidean space. The asymptotic cone of any bounded-diameter space is a point.
Behrstock, Drutu and Sapir look at the asymptotic cone of the mapping class group of a surface. One does this by choosing a finite generating set for this group, then constructing the Cayley graph for the group and the generating set, then setting each edge length equal to one to make the Cayley graph a metric space. The resulting space is somehow very close to being delta-hyperbolic (it’s related to the complex of curves, which is in fact delta hyperbolic) but it it’s not quite delta hyperbolic. It has these large Euclidean subspaces that come, for example, from taking Dehn twists along a collection of disjoint essential simples closed curves in the surface. These Dehn twists commute with each other so the subgroup of the mapping class group is Abelian. In the geometric picture, an Abelian group is Euclidean (since it is of the form ) and triangles in this Euclidean subspace are not delta thin. (Update: See Jason Behrstock’s comment for more on this.) But the philosophy is that if you find a way to ignore this flat regions, the space looks delta-hyperbolic.
Since the geometry of the mapping class group looks delta hyperbolic when you ignore the Euclidean parts, one would expect the asymptotic cone to combine tree-like features with Euclidean features. Behrstock, Drutu and Sapir show that there is a Lipschitz map from the asymptotic cone of the mapping class group into a direct product of trees. So, while the geometry doesn’t necessarily look like such a product, the topology of the asymptotic cone does. The direct product accounts for the large flat regions in the group. while the trees account for the delta-hyperbolic remainder.