Timeline for A discrete presentation for a free prop-$p$-group
Current License: CC BY-SA 3.0
8 events
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| Jun 21, 2015 at 12:53 | comment | added | Alireza Abdollahi | @YCor: No. I am not trying to compute the cohomology. I am interested in mathoverflow.net/questions/209438/… It is known that the number $R_G$ defined in the latter link is the minimum number of relations needed to present the finite $p$-group $G$ as a (topological) quotient of the free pro-$p$-group of rank $d_G$ the minimum number of generators of $G$. | |
| Jun 21, 2015 at 12:45 | comment | added | YCor | I'm not very optimistic that we can get anything beyond my suggestion and easy variants. Otherwise it sounds hopelessly hard. Are you trying to compute the cohomology of $F(n,p)$ as a discrete group? | |
| Jun 21, 2015 at 11:06 | comment | added | Alireza Abdollahi | @YCor: Is it possible to present the free pro-p-group as a free group quotient "without referring to the free pro-p-group itself"? Or maybe better to say, find a subset $X$ in $R_G$ which is minimal with respect to the inclusion such that $\langle \langle X \rangle \rangle=\langle \langle R_G \rangle \rangle$. Anyway, this may have a more or less trivial answer as the problem is to how define "explicit". | |
| Jun 21, 2015 at 10:46 | comment | added | YCor | I already did: for every group $G$ you consider the free group $F_G=\{e_g:g\in G\}$, the subset $R_G=\{e_{gh}e_h^{-1}e_g^{-1}:g,h\in G\}$, and you get $G=F_G/\langle\langle R_g\rangle\rangle$, where $\langle\langle X\rangle\rangle$ means the normal subgroup generated by $X$. | |
| Jun 21, 2015 at 10:43 | comment | added | Alireza Abdollahi | @YCor: Can you write down the presentation for $\mathbb{Z}_p$ in your way? | |
| Jun 21, 2015 at 10:34 | history | edited | Alireza Abdollahi | CC BY-SA 3.0 |
edited title
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| Jun 21, 2015 at 10:26 | comment | added | YCor | For the question itself, you have the canonical choice (as in any group), the free group over $\mathfrak{F}(n,p)$, with all the length three relations as relators. | |
| Jun 21, 2015 at 9:25 | history | asked | Alireza Abdollahi | CC BY-SA 3.0 |