MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $P_n$ denote the pro-$p$ completion of $F_n$ the free group of rank $n$. Given a (abstract) group homomorphism
$$ \phi:P_n\rightarrow G $$
where $G$ is a discrete group. Is $\phi$ continuous?

The case $G$ is finite is a Theorem of Serre and $\phi$ is continuous in this case.

share|cite|improve this question
up vote 6 down vote accepted

Not in general. Consider for instance the case that $G$ is the same abstract group as $P_n$, but with the discrete topology.

If $G$ is finitely generated, the answer is yes. See this article of Nikolov and Segal:

share|cite|improve this answer
Thank you very much for your answer and also the reference. – user19409 Mar 27 '13 at 23:43

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.