Let $G$ be a countable (that is edit) residually-$p$ group and let $\hat{G}\_p$ be its pro-$p$ completion.


1. If $\hat{G}\_p$ is finitely generated does it mean that $G$ is finitely generated?
2. If $\hat{G}\_p$ is finitely presented does it mean that $G$ is finitely presented? (I think the Grigorchuk group is not finitely presented, what about its pro-$p$ completion?)


**Edit:** Following @Ycor comment. Let me add that in Question 2, I would like $G$ to be finitely generatd. Also, let me be explicit:


3. Is the pro-$p$ completion of the Grigorchuk group finitely presented?



Finitely generated and finitely presented groups and their pro-$p$ completions
As I said in a comment, the answer to 1 is trivially no.


The answer to 2 (as edited) is also no. Platonov and Tavgen produced a finitely generated, infinitely presented subgroup $H$ in the square $F\times F$ of a free groups such that the inclusion induces an isomorphism of profinite completions (and hence of pro-$p$-completions for all $p$).


Here, $F$ can be constructed as follows. Consider any epimorphism $p:F\to P$ from $F$ to an infinite finitely presented group $P$ with no nontrivial finite quotient; then $H$ is defined as the fibre product $\{(g,h)\in F\times F:p(g)=p(h)\}$. That $P$ is finitely presented is used to ensure that $H$ is finitely generated. That $P$ has no nontrivial proper quotient is used by Platonov and Tavgen to prove that the inclusion induces an isomorphism of profinite completions. That the fibre product $H$ is infinitely presented is a classical fact on fiber products.


Reference: V. Platonov O.Tavgen.
On the Grothendieck problem of profinite completions of groups. (Russian) 
Dokl. Akad. Nauk SSSR 288 (1986), no. 5, 1054-1058. English translation: Soviet Math. Dokl. 33 (1986), no. 3, 822-825. [(MR link)](https://mathscinet.ams.org/mathscinet-getitem?mr=852649)


