My research area is mainly pro-$p$ groups and profinite groups. However, in the last few year I became also interested in discrete groups. Therefore, it seems to me a natural problem to look for examples of finitely generated groups such that their profinite completion is a pro-$p$ group. One trivial example is when the profinite completion is trivial or a finite $p$-group, e.g. an infinite simple group. Hence, it makes sense to require the group to be infinite and residually finite. But then you can take residually finite, $p$-groups, e.g., the Grigorchuk group. In such case tha profinite completion is pro-$p$ for an easy reason. 


So here are my questions:


1. Is there an infinite finitely generated residually finite non-torsion group such that its profinite completion is pro-$p$?
2. Is there an infinite finitely generated residually finite torsion free group such that its profinite completion is pro-$p$?


(Notice that profinite completion being pro-$p$ means that all subgroups of finite index are of $p$-power index.) 



When is the profinite completion a pro-$p$ group?
Gustavo A. Fernández-Alcober, Alejandra Garrido and Jone Uria-Albizuri gave a positive answer to question 2 and therefore also to question 1 in [On the congruence subgroup property for GGS-groups](http://www.ams.org/journals/proc/2017-145-08/S0002-9939-2017-13499-6/). 


