I call a profinite group $G$ ***Noetherian***, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is finitely generated.


A profinite group $G$ is called ***just-infinite*** if every nontrivial $M \lhd\_c G$ is open.


Let $K$ be a profinite Noetherian just-infinite group. Must $K$ be the profinite completion of some residually finite group $R$?



Profinite completions
The answer is **yes** in general.


Since $K$ is finitely generated, by the Nikolov-Segal theorem it coincides with its own profinite completion. So you simply may take $R=K$.


Perhaps, you also want $R$ to be finitely generated. In this case the answer is **no**. 


If $K$ is a non-soluble p-adic analytic pro-$p$ group, then it has polynomial subgroup growth. However, finitely generated groups of polynomial subgroup growth are virtually soluble (see the book "Subgroup Growth" of Lubotzky and Segal).


