Timeline for Amenability of Thompson's group looking at a 4-manifold having it as the fundamental group
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2011 at 21:35 | comment | added | user6976 | a-T-menability is a strong negation of property (T). The class of a-T-memable groups contains all amenable groups. Another approximation of amenability is G.Yu's property A. It is not known whether $F$ satisfies property A. That question seems to be as hard as amenability. Free groups and even all hyperbolic groups satisfy A. | |
| Oct 27, 2011 at 21:26 | comment | added | Valerio Capraro | Well, also free groups are a-T-menable... this is, by the way and little off-topic, one of the reasons why I don't understand very well in which sense a-T-menability is considered close to amenability. Anyway, many thanks for the information. | |
| Oct 27, 2011 at 20:17 | comment | added | user6976 | You can deduce that $F$ is a-T-menable, for example. That is the closest to amenability known property of $F$. | |
| Oct 27, 2011 at 20:13 | vote | accept | Valerio Capraro | ||
| Oct 27, 2011 at 20:12 | comment | added | Valerio Capraro | OK, thank you. What kind of information one can get in this other way; namely viewing $F$ as a fundamental group of an infinite dimensional cubical complex with $CAT(0)$ universal cover? | |
| Oct 27, 2011 at 19:59 | comment | added | user6976 | About 2), the construction of Kuzmin is explicit. About 3): there were no attempts (successful or not) because the idea does not seem to be fruitful: there seems to be no info you can get from the manifold that you cannot obtain just by looking at the very simple defining relations of $F$. One can get more information viewing $F$ as a fundamental group of an infinite dimensional cubical complex with CAT(0) universal cover. | |
| Oct 27, 2011 at 19:51 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
added 130 characters in body
|
| Oct 27, 2011 at 19:41 | history | edited | Valerio Capraro | CC BY-SA 3.0 |
added 416 characters in body; added 9 characters in body
|
| Oct 27, 2011 at 19:32 | comment | added | Valerio Capraro | I was just thinking, while having a shower, to modify the question! | |
| Oct 27, 2011 at 19:31 | comment | added | Mariano Suárez-Álvarez | @Valerio: it usually works better if you ask the subquestions explicitly :) | |
| Oct 27, 2011 at 19:28 | answer | added | user6976 | timeline score: 8 | |
| Oct 27, 2011 at 19:18 | comment | added | Valerio Capraro | Indeed my question contains many (maybe trivial) subquestions: 1) Does a 4-manifold with amenable fundamental group have any characterizing properties? 2) Is there an explicit way to construct a 4-manifold whose fundamental group is the Thompson group? | |
| Oct 27, 2011 at 18:49 | comment | added | Andy Putman | Since every group is the fundamental group of a $4$-manifold but most groups are not amenable, I can't see how this would buy you anything. One would need a $4$-manifold with some kind of extra structure that not all $4$-manifolds have, and I don't see how to do that with Thompson's group. | |
| Oct 27, 2011 at 18:24 | history | asked | Valerio Capraro | CC BY-SA 3.0 |