Timeline for A group, neither amenable, nor having a subgroup that looks like $F_2$ up to level $n$?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 11, 2017 at 8:49 | comment | added | YCor | OK, so it's indeed very confusing, especially in view of the informal definition "looks like $F_2$ up to level $n$". | |
| Nov 11, 2017 at 3:15 | comment | added | Bjørn Kjos-Hanssen | @YCor yes I did mean positive words, although it's of interest either way | |
| Nov 11, 2017 at 0:20 | comment | added | YCor | By the way I interpreted "words" as "group words". If the OP really means positive words (so "unfree" would be a misleading terminology), my above argument is still valid, but the converse probably fails (e.g., I guess there is no law of the form $uv^{-1}$ with $u,v$ positive words (possibly null), satisfied by all metabelian groups). | |
| Nov 11, 2017 at 0:12 | comment | added | YCor | @LucGuyot index nontrivial elements of the 2n-ball in $F_2=\langle x,y\rangle$ as $w_0,\dots,w_m$ with $w_0=x$, $w_1=y$. Define by induction $c_0=x$, $c_n=[w_n,c_{n-1}]$ if this $\neq 1$, and otherwise $c_n=[w_n,[w_{n-1},c_{n-1}]]$. Here I use hat $c_{n-1}$ has the form $[w_{n-1},c_{n-1}]$ and is $\neq 1$ and hence does not commute with $w_{n-1}$, and $w_n$ can't commute with both $c_{n-1}$ and $[w_{n-1},c_{n-1}]$. Set $c=c_m$. Then in any $n$-unfree group, for all $a,b$ there's $i$ such that $w_i(a,b)=1$ and hence $c(a,b)=1$. | |
| Nov 10, 2017 at 23:42 | comment | added | Luc Guyot | @YCor It may be obvious, but I don't see why the equivalence holds. | |
| Nov 7, 2017 at 8:39 | comment | added | YCor | To be length-$n$ unfree for some $n$ is equivalent to the better known notion of satisfying a group law (or "group identity") | |
| Nov 7, 2017 at 6:36 | comment | added | Bjørn Kjos-Hanssen | @LSpice thanks, I changed the title | |
| Nov 7, 2017 at 6:35 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |
better title, as @LSpice asked for
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| Nov 7, 2017 at 3:15 | vote | accept | Bjørn Kjos-Hanssen | ||
| Nov 7, 2017 at 3:09 | answer | added | user6976 | timeline score: 12 | |
| Nov 7, 2017 at 2:59 | comment | added | LSpice | I'm sure I'm being silly, but your title asks for a group that "looks like $\mathrm F_2$ up to level $n$", whereas your question asks for a group that "is level-$n$ unfree". These seem like opposite goals. | |
| Nov 7, 2017 at 1:41 | history | asked | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |