# Faithful representations of free pro-p groups

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 $K$.

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:

• 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. – Ian Agol Aug 6 '14 at 15:44
• 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). – Joël Aug 6 '14 at 16:06