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Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F \rightarrow \text{GL}_n(\overline{\mathbb{Q}_p})$?

Note that every compact subgroup of $\text{GL}_n(\overline{\mathbb{Q}_p})$ is conjugate to a subgroup of $\text{GL}_n(\mathcal{O}_K)$ where $K$ is a finite extension of $\mathbb{Q}_p$ and $\mathcal{O}_K$ is the integral closure of $\mathbb{Z}_p$ in $K$.

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It will only happen if $m=1$. See this paper:

http://mlarsen.math.indiana.edu/~larsen/papers/2gen.pdf

Indeed, the pro-$p$ groups that are linear over local fields of characteristic $0$ are just the pro-$p$ groups of finite subgroup rank.

Edit: As Ian Agol suggested, you should look at 'Analytic pro-p Groups' by Dixon, de Sautoy, Mann and Segal:

books.google.com/books?id=Fjq-ngEACAAJ

It develops a remarkably powerful theory (sorry about the pun) for these groups and is well worth looking at if you are interested in linear groups over the p-adics (and extensions thereof).

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  • $\begingroup$ Great! but this is stated in the article without proof. Do you have any reference for this claim (finite subgroup rank)/ simple argument? $\endgroup$
    – Pablo
    Aug 6, 2014 at 15:22
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    $\begingroup$ 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. $\endgroup$
    – Ian Agol
    Aug 6, 2014 at 15:44
  • $\begingroup$ But why analytic implies finite subgroup rank? $\endgroup$
    – Pablo
    Aug 6, 2014 at 16:03
  • $\begingroup$ 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). $\endgroup$
    – Joël
    Aug 6, 2014 at 16:06
  • $\begingroup$ 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. $\endgroup$
    – Pablo
    Aug 6, 2014 at 16:09

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