Profinite Topologies on Free Products of Groups
by
Pavel Zalesskii
University of Brasilia
Coauthors: L. Ribes (Carleton University)

Let C be an extension closed pseudovariety of finite groups. Let G be a group and N_C the collection of all nomal subgroups N of G such that the quotient G/N belongs to C. Then there is a unique topology on G making it into a topological group such that N_C is a fundamental system of neighborhoods of the identity element 1 of G. This topology is called the pro-C topology of G.

A group G is said to be 2-product separable (with respect to its pro-C topology) if the product HK of any two finitely generated closed subgroups H and K of G is closed in the pro-C topology of G.

Theorem.  Let C be an extension closed pseudovariety of finite groups. Let G be a free product of 2-product separable groups (with respect to the pro-C topology). Then G is 2-product separable.

When C is the pseudovariety of all finite groups the result was proved by T. Coulbois, Free products, profinite topology and finitely generated subgroups, Internat. J. Algebra Comput., 11 (2001) 171-184 for the product of an arbitrary number of subgroups.

Date received: December 29, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-16.