Timeline for Følner sequences with weird shapes
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jan 15, 2021 at 16:59 | answer | added | ARG | timeline score: 1 | |
| Jan 14, 2021 at 11:29 | comment | added | ARG | @DiegoMartínez I kind of agree but "rectangular" only make sense if your group splits:following the proof that a semi-direct prodcut of amenable groups is amenable, you get "rectangular"sets.But I'm not sure if this is true for non-split extensions(can't remember good examples).And this surely can't be true for simple amenable groups (because "rectangular" does make sense there). Lastly, it's just one of many sequences,e.g. in nilpotent groups you have "rectangular"and "round"Folner sequences.It looks like[to me]that the "rectangular"option is the one that comes out of the box. | |
| Jan 14, 2021 at 11:16 | comment | added | Diego Martinez | @ARG your post and this discussion does raise the question of whether Følner sets may be assumed to be rectangular. I personally don't know any example when this does not happen (as it does in extensions). Even in some more intricate cases, as topological full groups of minimal $\mathbb{Z}$-actions on the Cantor space, Følner sets turn out to be rectangular. | |
| Jan 14, 2021 at 11:05 | comment | added | ARG | @DiegoMartínez Correct. I thought, that there might still be important informations in the linked question. For example: we don't know of any group of exponential growth where Folner sequences are balls. However, if $G$ and $H$ are amenable, then one can show that $G \rtimes H$ is amenable and that Folner sets are of the Form $E \times F$. So you should rather expect "rectangular" Folner sets (whenever that makes sense) than ball-like Folner sets. This becomes more obvious when you look at the optimal Folner sets. Because, even in $\mathbb{Z}^d$ optimal Folner sets are "rectangular". | |
| Jan 14, 2021 at 10:55 | comment | added | Diego Martinez | @ARG your points are appreciated. You are indeed right that the rectangles I mention in the post are star-shaped (as they should), but morally speaking not every star-shaped set needs be a rectangle (whatever that means). I didn't mention optimal Folner sets as I'm not really interested in whether they are optimal, only whether they are balls, but the optimality does indeed get rid of some technicalities the question does not get into. | |
| Jan 14, 2021 at 8:33 | comment | added | ARG | The problem with optimal sets is that there is essentially no group where it's proven what they are. For example, it's an long-standing open question to prove what are such sets in the (continuous) Heisenberg group (although the conjectured shape is well-described). That was my motivation for this question. Optimal sets have the fun property that they have a relatively good Cheeger constant (but it can be much lower than their isoperimetric ratio see here ). | |
| Jan 14, 2021 at 8:23 | comment | added | ARG | @DiegoMartínez two comments: (1) the "rectangles" you mention are actually "star-shaped" in the sense of my question,... so those two do not really go in opposition. (2) it can be useful to reduce the arbitrarity in the Folner set by restricting to "optimal" Folner sets. Definition: let $I(n) = inf |\partial A|/|A|$ (where the inf runs over all sets $A$ of size $\leq n$) be the isoperimetric profile. Then a set $F$ is optimal if $I(|F|) = |\partial F|/|F|$. If you look at such sets then they probably are not "strange" at all (since they must be extremely wellchosen).But they sure are no balls. | |
| Jan 14, 2021 at 7:38 | answer | added | markvs | timeline score: 2 | |
| Jan 13, 2021 at 12:12 | history | edited | YCor |
edited tags; edited tags
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| Jan 13, 2021 at 11:55 | comment | added | Diego Martinez | @YCor even if the question is not well-posed (which it isn't, as you rightly claim), I think it still makes sense, and your and Ville Salo are answering it quite well. I was looking for a list of groups where balls are not Folner sets, but these are known (and explicit). I could delete the question if needs be, but I think this might be useful for somebody later. | |
| Jan 13, 2021 at 11:45 | comment | added | YCor | In a sense I don't think this is the right question. In amenable groups of exponential growth balls are seldom/never Følner (even after extraction). Also perturbing Følner $(F_n)$ by subsets of size $o(F_n)$ yields Følner, so one should phrase the question in a way to ignore this. On the other hand, in polycyclic groups or lamplighter groups, one can always find Følner subsets $(F_n)$ that are trapped between the $cn$-ball and the $Cn$-ball (see arxiv.org/abs/math/0603138). This behavior is the most "regular" one, and doesn't qualify as "weird". | |
| Jan 13, 2021 at 11:27 | answer | added | Ville Salo | timeline score: 3 | |
| Jan 13, 2021 at 10:28 | comment | added | Ville Salo | Fair enough. I have not talked to everyone! | |
| Jan 13, 2021 at 10:23 | comment | added | Diego Martinez | @VilleSalo I've heard more than once that Følner sets are usually strange, but this is of course personal. I do agree with you on the lamplighter, and it fits the question perfectly. | |
| Jan 13, 2021 at 10:22 | comment | added | Diego Martinez | @YCor you are indeed right that "weird" is subjective. I just meant Folner sequences that are not balls. Hence your example of $\mathbb{Z}$ fits the question, but I'd like some other examples. | |
| Jan 13, 2021 at 10:13 | comment | added | Ville Salo | I do disagree with the statement that balls are natural shapes. For example on the lamplighter group, balls look pretty strange, while Fölner sequences are beautiful. | |
| Jan 13, 2021 at 10:07 | comment | added | Ville Salo | I'm not sure I agree with the premise of the question though, I have more often (= at least once) heard "there is a unique choice for the Fölner sequence", which does not imply they are not strange, but does imply they are natural (I guess this statement refers to mutual tileability). | |
| Jan 13, 2021 at 10:07 | comment | added | YCor | It's quite subjective... you could have a "weird" subsequence of "weird"-shaped balls. Also on $\mathbf{Z}$ the union of $\{1,\dots,n\}$ with any "weird" subset of cardinal $o(n)$ is Følner. Also in polycyclic groups or the $ax+b$ group the rectangles you mention look all but weird for me. The exponential size is with respect to matrix coordinates, but it's of linear size in the word length. | |
| Jan 13, 2021 at 10:05 | comment | added | Ville Salo | Maybe Fölner sequences of group extensions fit the bill, see e.g. terrytao.wordpress.com/2009/04/14/some-notes-on-amenability | |
| Jan 13, 2021 at 9:59 | history | asked | Diego Martinez | CC BY-SA 4.0 |