Timeline for Amenability with respect to a function
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 12, 2011 at 15:02 | vote | accept | Valerio Capraro | ||
| Sep 11, 2011 at 0:52 | answer | added | Andreas Thom | timeline score: 2 | |
| Sep 5, 2011 at 6:21 | comment | added | Valerio Capraro | Well, take a copy of $\mathbb Z$ inside $\mathbb F_2$ and define $\mu$ on $\mathbb F_2$ to be the trivial extension of an invariant mean on this copy of $\mathbb Z$. This is a measure on $\mathbb F_2$ which is null on every finite set. It follows that it is OK for all finitely supported $\phi$. Doesn't it? | |
| Sep 4, 2011 at 23:32 | comment | added | user6976 | I think you should try to consider the indicator case yourself. It should be an easy exercise. Then you will get an idea what to do next. | |
| Sep 4, 2011 at 20:30 | comment | added | Valerio Capraro | Thanks for the reference. Do you know the answer about the indicator function of a generator? I can't see any new real information respect to the indicator function of a point or even repect to a finitely supported function. | |
| Sep 4, 2011 at 17:49 | comment | added | user6976 | There are ergodic theorems for free group actions (Bufetov, Alexander I. Convergence of spherical averages for actions of free groups. Ann. of Math. (2) 155 (2002), no. 3, 929–944.) Perhaps that can help. But I would suggest that you ask the simplest question for which you do not know the answer first. How about the indicator function of a generator? | |
| Sep 4, 2011 at 16:18 | comment | added | Valerio Capraro | No.. Actually I don't know the answer with $\mathbb F_2$. I have some trouble even when $\phi$ is finitely supported. In this case $\mu$ should be such that every finite set has zero measure, but I am not able to contruct (ot to prove the existence) of such a measure in a non-amenable group. | |
| Sep 4, 2011 at 15:57 | comment | added | user6976 | Did you try any $\phi$ at all? | |
| Sep 4, 2011 at 15:28 | comment | added | Valerio Capraro | I have no idea. Is it trivial that for any $\phi$ there is such a measure? | |
| Sep 4, 2011 at 15:05 | comment | added | user6976 | How about the free group with 2 generators? | |
| Sep 4, 2011 at 14:45 | history | asked | Valerio Capraro | CC BY-SA 3.0 |