Assume that we have two residually finite groups $G$ and $H$. Which properties of $G$ and $H$ could be used to show that their pro-finite (or pro-p) completions are different?


I asked a while ago in the group-pub mailing list whether finite presentability is such a property but Lubotzky pointed out that it is not the case. A finitely presented and an infinitely presented group can have isomorphic pro-finite completions.



Distinguishing pro-finite completions
There's a theorem that two finitely generated residually finite groups have the same profinite completions if and only if they have the same finite quotients. A reference for the statement of this is [Theorem 2 of this paper](http://www.ma.utexas.edu/users/areid/Groth_revised.pdf), but they cite [Ribes and Zalesskii](http://books.google.com/books?id=47ouE_XSJZYC&lpg=PP1&dq=ribes%2520zalesskii&pg=PP1#v=onepage&q&f=false) for the proof. 


