Profinite Topologies in Free Products of Groups

Let H be an abstract group and let C be a variety of finite groups (i.e., a class of finite groups closed under taking subgroups, quotients and finite direct products); for example the variety of all finite p-groups, for a fixed prime p. Consider the smallest topology on H such that all the homomorphism H −→ C from H to any group C ∈ C (endowed with the discrete topology) is continuous. We refer to this topology as the pro-C topology of H. This paper is concerned with the following property on H: whenever H1 and H2 are finitely generated subgroups of H such that H1 and H2 are closed in the pro-C topology of H, then the subset H1H2 of H is closed. If H has this property, we call H “2-product subgroup separable” (relative to the class C; there is an analogous concept of “n-product subgroup separable”). The original motivation for the study of this property 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 (see [5], [6]). For example, if C is in addition closed under extensions, then groups that are extensions of free groups by groups in C are n-product subgroup separable, for any natural number n (see [8], [9]; see also [12] for other examples). In this paper we show that if the variety C is closed under extensions, then the property of being 2-product subgroup separable is preserved by taking free products of groups (see Theorem 3.13). This extends in one direction an analogous result of T. Coulbois [1]. The methods used to prove this result are based in the theories of groups acting on trees and of profinite groups acting on profinite trees.