Timeline for When are Carnot groups negatively curved and homeomorphic to Euclidean space
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 7, 2019 at 10:12 | vote | accept | ABIM | ||
| May 2, 2019 at 9:06 | history | edited | YCor | CC BY-SA 4.0 |
edited title
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| May 2, 2019 at 7:47 | answer | added | YCor | timeline score: 3 | |
| May 2, 2019 at 3:34 | comment | added | Anton Petrunin | @YCor I think it is better to make an answer from your comment, so the question will not appear in "unanswered". | |
| May 1, 2019 at 14:39 | comment | added | ABIM | Amazing, if you want to post the exact same comment as an answer I'd be more than happy to accept. Also, I'll try out the exercise :) | |
| May 1, 2019 at 12:52 | history | edited | YCor | CC BY-SA 4.0 |
edited title
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| May 1, 2019 at 12:13 | comment | added | YCor | PS I don't know if it's part of the question but all Carnot groups are complete, since they have all closed balls compact. In general, any metric space with a transitive isometry group, and having a compact subset with nonempty interior, is complete (easy exercise). | |
| May 1, 2019 at 12:12 | comment | added | YCor | Scott D. Pauls. The large scale geometry in nilpotent Lie groups. Commun. Anal. Geom. 9(5), 951-982, 2001. However, the result that every Carnot group of dimension $\ge 2$ is not CAT($-\kappa$) for $\kappa>0$ is straightforward. Indeed, since it has a non-isometric self-homothety, it would imply that it is CAT($-\kappa'$) for every $\kappa'>0$, and hence CAT($-\infty$), which for a geodesic space means an $\mathbf{R}$-tree, which cannot have any subset homeomorphic to the plane. | |
| May 1, 2019 at 12:05 | comment | added | ABIM | Do you have a reference to this paper, by any chance? | |
| May 1, 2019 at 11:47 | comment | added | YCor | Abelian Carnot groups are isometric to Euclidean spaces. So no, they're not negatively curved (in dimension $\ge 2$) although they're non-positively curved. | |
| May 1, 2019 at 11:46 | comment | added | ABIM | Interesting! This i did not know. | |
| May 1, 2019 at 11:43 | comment | added | YCor | Only abelian ones. It's a theorem of Pauls that a nonabelian simply connected nilpotent groups can't even QI embed into any CAT(0) space. | |
| May 1, 2019 at 11:33 | history | asked | ABIM | CC BY-SA 4.0 |