FREE PRODUCT, PROFINITE TOPOLOGY AND FINITELY GENERATED SUBGROUPS

We consider the following property for a group G:(RZn)ifH1,…,Hnare finitely generated subgroups of G then the setH1 H2⋯ Hn= {h1 ⋯ hn| h1∈ H1, …,hn∈ Hn}is closed with respect to the profinite topology of G. It is obvious that finite groups and finitely generated commutative groups have the property ( RZ n). L. Ribes and P. Zalesskiĭ proved that any free group has ( RZ n). We show that the property ( RZ n) is stable under the free product operation. We use techniques developed by B. Herwig and D. Lascar on the one hand, R. Gitik on the other hand.