The original motivation for this work goes back to a problem posed by J. Rhodes on the existence of an algorithm to compute the so called kernel of a finite monoid. This gave rise to studies by L. Ribes and P. Zalesskii of profinite topologies on a free group, providing an approach to answer that problem and related ones posed by J.-É. Pin. These topological results motivated work by B. Herwig and D. Lascar who, using model theoretic methods, proved some of the Ribes-Zalesskii results. In turn, methods similar to those of Herwig-Lascar have been employed by T. Coulbois to study the profinite topology of free products of groups.
The work that we present here (joint with P. Zalesskii) is an attempt to generalize the results of T. Coulbois. Let be an extension closed variety of finite groups (a nonempty class of finite groups closed under taking subgroups, homomorphic images and extensions); for example the class of all finite groups or the class of all finite -groups, for a fixed prime . Given an abstract group , its pro- topology is determined by imposing that a fundamental system of neighboorhoods of consists of the normal subgroups of such that . Given a natural number , we say that an abstract group is -product subgroup separable (with respect to ) if whenever are finitely generated subgroups of which are closed in the pro- topology of , then the set is closed. We prove the following
Theorem A finite free product of groups that are -product subgroup separable is -product subgroup separable.
The methods used to prove this results exploit the theories of groups acting on trees and of profinite groups acting on profinite trees.
Sponsored in part by the FCT approved projects POCTI 32817/99 and POCTI/MAT/37670/2001 in participation with the European Community Fund FEDER and by FCT through Centro de Matemática da Universidade do Porto. Also sponsored in part by FCT, the Faculdade de Ciências da Universidade do Porto, Programa Operacional Ciência, Tecnologia, Inovação do Quadro Comunitário de Apoio III, and by Caixa Geral de Depósitos.