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 :)

Mike

]]>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.

]]>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.

]]>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Groves has an as-yet-unpublished result that applies to give both sets of results (on pure pseudo-Anosov subgroups and property T subgroups).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.

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