Regularity of asymptotic cones - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:37:50Z http://mathoverflow.net/feeds/question/46030 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46030/regularity-of-asymptotic-cones Regularity of asymptotic cones Marcin Kotowski 2010-11-14T10:11:56Z 2010-11-15T00:07:22Z <p>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?</p> http://mathoverflow.net/questions/46030/regularity-of-asymptotic-cones/46038#46038 Answer by Simon Thomas for Regularity of asymptotic cones Simon Thomas 2010-11-14T12:59:19Z 2010-11-14T12:59:19Z <p>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.</p> http://mathoverflow.net/questions/46030/regularity-of-asymptotic-cones/46089#46089 Answer by Alessandro Sisto for Regularity of asymptotic cones Alessandro Sisto 2010-11-15T00:07:22Z 2010-11-15T00:07:22Z <p>I would just like to add to the answer by Simon Thomas that</p> <p>-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.</p> <p>-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 <code>\$2^{\aleph_0}\$</code> 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).</p>