Timeline for An infinite profinite group such that any $\overline{\mathbb{F}_{p}((t))}$-adic representation has finite image
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jun 12, 2022 at 8:43 | comment | added | YCor | For every prime $p$ and every $n\ge 2$, the free pro-$p$-group on $n$ generators is not, as topological group, linear over any local field (Barnea-Larsen J. Algebra 1999 DOI link) | |
| Jun 11, 2022 at 23:54 | comment | added | Will Sawin | I think the quotient of the free rank $n$ pro-$p$ group by $n$ elements chosen at random should have this property with probability 1, but I don't know if this is a theorem. | |
| Jun 11, 2022 at 22:16 | comment | added | YCor | Yes, because for $n\ge 3$, the group $\mathrm{SL}_n(\mathbf{Z})$ has no faithful finite-dim representation over any field of finite characteristic. This is because there is no distorted infinite cyclic subgroup in these groups. This also apply to many other groups for the same reason, e.g., the profinite or pro-$p$-completion of the integral Heisenberg group. This is also true for $n=2$, arguing with $\mathrm{SL}_2(\mathbf{Z}[1/\ell])$ for any choice of prime $\ell\neq p$. | |
| Jun 11, 2022 at 22:03 | history | asked | Nobody | CC BY-SA 4.0 |