## Regularity of asymptotic cones

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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?

-

## 2 Answers

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.

-

I have a preprint about that problem: arxiv.org/abs/1010.1199 – Alessandro Sisto Nov 14 2010 at 23:38
@ Alessandro: your preprint looks very interesting! – Simon Thomas Nov 15 2010 at 1:00

### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.

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).

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9278703927993774, "perplexity_flag": "head"}