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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.

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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: http://arxiv.org/abs/1102.3037

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

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