A constructive version of the Ribes-Zalesski product theorem

Abstract

For any given finitely generated subgroups H1,...,Hn of a free group F and any element w of F not contained in the product H1⋯Hn, a finite quotient of F is explicitly constructed which separates the element w from the set H1⋯Hn. This provides a constructive version of the “product theorem”, stating that H1⋯Hn is closed in the profinite topology of F. The method of proof also applies to other profinite topologies. It is efficient for the profinite topology as well as for the pro-p topology of F. The main tools used are universal p-extensions and inverse automata.