Timeline for Equations in finite subgroups of unitary groups
Current License: CC BY-SA 3.0
18 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Nov 26, 2016 at 11:47 | comment | added | Andreas Thom | @BjørnKjos-Hanssen, any such equality $x=y$ would be a semigroup identity for $Sym(n)$. I do not think that those can be shorter than $C\exp(n)$, but I am not completely sure. | |
| Nov 26, 2016 at 1:31 | comment | added | Bjørn Kjos-Hanssen | @AndreasThom I wonder if these words $w$ with $w(u,v)=1_n$ can be chosen so that $w(u,v)=1_n$ is equivalent to $x=y$ for two words (with no inverses being taken) of the same length $x$ and $y$. | |
| Nov 25, 2016 at 22:21 | comment | added | Andreas Thom | I think I can prove what I claimed in a few pages. | |
| Nov 25, 2016 at 22:10 | comment | added | Andreas Thom | The shortest known identity for $Sym(n)$ is of length $\exp(C \log(n)^4 \log(\log(n)))$. I would expect that this is the crucial case, which dominates the estimates for the length of laws for all finite subgroups. There is no non-trivial known (at least to me) lower bound for the length of such an identity (be it for $Sym(n)$ or all finite subgroups of $U(n)$). The trivial lower bound is linear. | |
| Nov 25, 2016 at 19:10 | comment | added | YCor | So the question can be formulated as follows: let $K_n$ be the intersection of all kernels of all homomorphisms from the free group $F_2=<x,y>$ to finite subgroups of $U(n)$. Let $q_n$ be the smallest size of a nontrivial element in $K_n$. Can we have good estimates on $q_n$? (One can probably find elements in $K_n$ of the form $[x^m,y^m]$ with $m$ exponentially large w.r.t $n$, according to the previous discussion). In scholar language, $F_2/K_n$ is the free group in the variety generated by finite subgroups of $U(n)$, and the question is about its girth. | |
| Nov 25, 2016 at 17:09 | comment | added | Bjørn Kjos-Hanssen | @AndreasThom Oh yes that would help, see updated version of question. | |
| Nov 25, 2016 at 17:06 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |
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| Nov 25, 2016 at 10:22 | comment | added | Andreas Thom | I would guess that something on the order $O(\exp(n))$ is enough (and tight). However, if it helps, there should be much shorter non-trivial words $w \in F_2$, such that $w(u,v)=1_n$ whenever $u,v$ generate a finite group. | |
| Nov 25, 2016 at 7:59 | history | edited | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |
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| Nov 25, 2016 at 7:55 | history | undeleted | Bjørn Kjos-Hanssen | ||
| Nov 25, 2016 at 7:53 | history | deleted | Bjørn Kjos-Hanssen | via Vote | |
| Nov 25, 2016 at 4:40 | comment | added | YCor | PS these provide upper bounds rather close to $n!$. Here the worse case could be closer to the exponent of the symmetric group (lcm of first $n$ integers), which seems to rather grow exponentially. | |
| Nov 25, 2016 at 3:22 | comment | added | YCor | OK, it's closely related anyway: first you consider $U(n)$ while the other question also considers $O(n)$. Second, you specify to a special case (two generators and their powers). Thom's answer to the linked question provides you a reasonable upper bound on $m_n$. | |
| Nov 25, 2016 at 3:01 | review | Close votes | |||
| Nov 25, 2016 at 4:45 | |||||
| Nov 25, 2016 at 2:45 | comment | added | YCor | Possible duplicate of Bound on the index of an abelian subgroup in discrete subgroup of the euclidean group? | |
| Nov 25, 2016 at 2:43 | comment | added | YCor | I guess you mean, for given $n$, what is the smallest $m_n\ge 1$ with this property? (Compute or at least estimate $m_n$?) | |
| Nov 25, 2016 at 0:22 | history | asked | Bjørn Kjos-Hanssen | CC BY-SA 3.0 |