Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $K$.

share|cite|improve this question

1 Answer 1

up vote 7 down vote accepted

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

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:

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).

share|cite|improve this answer
Great! but this is stated in the article without proof. Do you have any reference for this claim (finite subgroup rank)/ simple argument? – Pablo Aug 6 '14 at 15:22
You might have a look at "Analytic pro-p groups": 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
But why analytic implies finite subgroup rank? – Pablo Aug 6 '14 at 16:03
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
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. – Pablo Aug 6 '14 at 16:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.