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Are there any general conditions guaranteeing that the asymptotic cone of a group/graph is "regular" in some sense? E.g. for $\mathbb{Z}^d$ we get $\mathbb{R}^d$ as the asymptotic cone, which is even a manifold, but for general groups we only get a metric space without additional structure. Does knowing that asymptotic cone is regular (e.g. a manifold) imply any properties of the original group?

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Drutu has shown that if every asymptotic cone of the finitely generated group $G$ is a proper space, then $G$ has polynomial growth; and hence by Gromov's Theorem, it follows that $G$ is nilpotent-by-finite. It seems to be open whether or not the conclusion holds if just one asymptotic cone of $G$ is proper.

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I have a preprint about that problem: arxiv.org/abs/1010.1199 –  Alessandro Sisto Nov 14 '10 at 23:38
    
@ Alessandro: your preprint looks very interesting! –  Simon Thomas Nov 15 '10 at 1:00
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I would just like to add to the answer by Simon Thomas that

-if a group is virtually nilpotent, its asymptotic cones are very regular: they have a Lie group structure and their metric is of Carnot-Caratheodory type (these metrics are described in the wikipedia article "Sub-Riemannian manifold"). Also, the asymptotic cones do not depend on the scaling factor.

-if a group is not virtually nilpotent, its asymptotic cones tend to be VERY large objects. For example, the asymptotic cones of each non-virtually cyclic hyperbolic group are real trees with valency $2^{\aleph_0}$ at each point (those groups have exponential growth, I have to admit that I know very little about asymptotic cones of groups of intermediate growth).

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