THE PRODUCT SEPARABILITY OF THE GENERALIZED FREE PRODUCT OF CYCLIC GROUPS

Let G be a group endowed with its profinite topology, then G is called product separable if the profinite topology of G is Hausdorff and, wh enever H-1, H-2,..., H-n are finitely generated subgroups of G, then t he product subset H-1 H-2... H-n is closed in G. In this paper, we pro ve that if G = F x Z is the direct product of a free group and an infi nite cyclic group, then G is product separable. As a consequence, we o btain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separab le. These results generalize the theorems of M. Hall Jr. (who proved t he conclusion in the case of n = 1, [3]), and L. Ribes and P. Zalesski i (who proved the conclusion in the case of that G is a finite extensi on of a free group, [6]).