Low Dimensional Topology

November 4, 2008

Aymptotic cones of mapping class groups

Filed under: Curve complexes,Hyperbolic geometry,Mapping class groups — Jesse Johnson @ 9:03 am

Here’s a recent preprint that sounds pretty interesting by Behrstock, Drutu and Sapir [1].  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 d_s(x,y) = sd(x,y) 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 \mathbf{Z}^n) 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.


  1. As I understand it, the principal motivation for proving this is the application mentioned in the abstract: if a group G has Kazhdan’s property T then there are only finitely many conjugacy classes of homomorphisms from G into a mapping class group.

    Results like this have been in the air for a little while now. Dahmani and Fujiwara proved that there are only finitely many conjugacy classes of pure pseudo-Anosov subgroups isomorphic to a given one-ended finitely presented group (which Bowditch had already done for surface groups). Bowditch, Dahmani and Fujiwara all use the curve complex rather than the asymptotic cone.

    Groves has an as-yet-unpublished result that applies to give both sets of results (on pure pseudo-Anosov subgroups and property T subgroups). I think he uses the curve complex and not the asymptotic cone.

    I suppose I’m saying that the asymptotic cone is a nice gadget, but it doesn’t seem to be absolutely necessary for this sort of application.

    Comment by Henry Wilton — November 5, 2008 @ 9:40 am | Reply

  2. Sure, applications are nice, but I think their asymptotic cone result is pretty interesting on its own. My impression from reading the abstract was that the application was not the initial motivation, but a nice little bonus.

    Comment by Jesse Johnson — November 5, 2008 @ 9:44 pm | Reply

  3. You’re quite right – and I confess I hadn’t actually looked at the paper before I commented.

    I do like asymptotic cones, they’re fun – but they also seem philosophically problematic. Their definition uses the Axiom of Choice in a very heavy way, and in practice it always seems to turn out that you can get by without them. They seem more like a slick tool than objects to study in their own right.

    It’s just my personal bias, really. But the application is a genuinely important one.

    Comment by Henry Wilton — November 6, 2008 @ 9:52 pm | Reply

  4. It is true that our initial motivation to write this paper was that we had ideas for proving finiteness results for subgroups of the mapping class group. But, it turned out that as we developed our techniques for doing so, we established several results about the asymptotic cone which in the end we felt were as interesting as the group theory results.

    Incidentally, I also had a comment to make about the original post above. In the post it is written “In the geometric picture, an Abelian subgroup is Euclidean” this is true if the word “subgroup” is replaced by “group” as the asymptotic cone of Z^n is R^n (although with the \ell^1 metric, since the word metric on Z^n is the \ell^1 not the \ell^2 metric). On the other hand, in a finitely generated group G it need not be the case that the induced metric on a Z^n subgroup is quasi-isometric to the standard metric on Z^n (when the induced metric and the intrinsic metric are comparable one says that the subgroup is undistorted). Readers of this blog are (perhaps secretly) well aware of groups which contain a Z^2 subgroup whose metric is distorted…such examples are provided by the fundamental group of any 3-manifold modeled on SOL. One can fairly easily convince oneself that the obvious Z^2 subgroups of a SOL 3-manifold are distorted; it actually turns out that such a group doesn’t have any quasi-isometrically embedded Z^2s, this follows from an interesting result if Jose Burillo who showed that these groups have one dimensional asymptotic cones.

    Comment by jason behrstock — April 8, 2009 @ 2:08 pm | Reply

  5. Thanks for the clarification. To translate this into topology, the 3-manifolds with SOL geometry are torus bundles with pseudo-Anosov monodromy, so the Z^2 subgroup is the image of the fundamental group of a leaf in thus bundle.

    Comment by Jesse Johnson — April 16, 2009 @ 12:35 pm | Reply

  6. In the Sol case it really is Anosov!

    Comment by Henry Wilton — April 17, 2009 @ 12:15 pm | Reply

  7. New here, from Toronto, Canada

    Just a quick hello from as I’m new to the board. I’ve seen some interesting posts so far.

    To be honest I’m new to forums and computers in general :)


    Comment by Mikeharvey — May 13, 2010 @ 4:31 pm | Reply

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