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.



Quotients of Free pro-p groups
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>


