, Volume 250, Issue 2, pp 287-297
Date: 07 Jan 2005

A constructive version of the Ribes-Zalesski product theorem

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Abstract.

For any given finitely generated subgroups H 1,...,H n of a free group F and any element w of F not contained in the product H 1H n , a finite quotient of F is explicitly constructed which separates the element w from the set H 1H n . This provides a constructive version of the “product theorem”, stating that H 1H n 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.

The authors gratefully acknowledge support from INTAS project 99–1224. The second author was supported in part by NSERC and by the FCT and POCTI approved projects POCTI/32817/MAT/2000 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER.