Timeline for Faithful representations of free pro-p groups
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 6, 2014 at 23:26 | history | edited | Colin Reid | CC BY-SA 3.0 |
added 344 characters in body
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| Aug 6, 2014 at 16:12 | comment | added | Joël | Ah yes, you're right. Sorry. | |
| Aug 6, 2014 at 16:09 | comment | added | Pablo | Dear Joel, if you look inside the proof of the theorem you have just mentioned, you will see that the char 0 case is referenced to the book that Ian Agol told us about. | |
| Aug 6, 2014 at 16:06 | comment | added | Joël | Pablo, what is stated without proof? I don't see your exact result stated, but Theorem 1.1 is the same with $\overline{ \mathbb Q_p}$ replaced by a local field, and it is proved in the paper. To go from the case of a local field to your case, you just have to know, as you said yourselves in your question, that since $F(p,m)$ is compact its image is in $GL_n(F)$ for a local field $F$. Of course, the proof of Theorem 1.1 in the paper is not long (two paragraphs) but it is a proof all the same (using a result of Pink, whose proof you can find in the reference given). | |
| Aug 6, 2014 at 16:03 | comment | added | Pablo | But why analytic implies finite subgroup rank? | |
| Aug 6, 2014 at 15:44 | comment | added | Ian Agol | You might have a look at "Analytic pro-p groups": books.google.com/books?id=Fjq-ngEACAAJ Since $F(p,m)$ is compact, the image must lie in a compact subgroup of $GL(n,F)$, which (up to finite index) is conjugate into $GL(n,\mathbb{O}_F)$. As shown in the book, such groups are "p-powerful", which in particular implies that they are analytic and not free. | |
| Aug 6, 2014 at 15:22 | comment | added | Pablo | Great! but this is stated in the article without proof. Do you have any reference for this claim (finite subgroup rank)/ simple argument? | |
| Aug 6, 2014 at 15:14 | vote | accept | Pablo | ||
| Aug 6, 2014 at 15:05 | history | answered | Colin Reid | CC BY-SA 3.0 |