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Journal of the London Mathematical Society 1997 56(1):91-103; doi:10.1112/S0024610796004425
© 1997 by London Mathematical Society
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© The London Mathematical Society

The Product Separability of the Generalized Free Product of Cyclic Groups

Shihong You

Department of Mathematics and Statistics, Carleton University Ottawa, Ontario, Canada K1S 5B6

Received 30 October 1993. Revision received 3 March 1995.

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, whenever H1, H2, ..., Hn are finitely generated subgroups of G, then the product subset H1 H2 ... Hn is closed in G. In this paper, we prove that if G=FxZ is the direct product of a free group and an infinite cyclic group, then G is product separable. As a consequence, we obtain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separable. These results generalize the theorems of M. Hall Jr. (who proved the conclusion in the case of n=1, [3]), and L. Ribes and P. Zalesskii (who proved the conclusion in the case of that G is a finite extension of a free group, [6]).


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