The Ribes-Zalesskii property of some one relator groups

Abstract

. The profinite topology on any abstract group G , is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group G has the Ribes-Zalesskii property of rank k , or is RZ k with k a natural number, if any product H 1 H 2 ··· H k of finitely generated subgroups H 1 ,H 2 , ··· ,H k is closed in the profinite topology on G . And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ k for any natural number k . In this paper we characterize groups which are RZ 2 . Consequently, we obtain condition under which a free product with amalgamation of two RZ 2 groups is RZ 2 . After observing that the Baumslag-Solitar groups BS ( m,n ) are RZ 2 and clearly RZ if m = n , we establish some suitable properties on the RZ 2 property for the case when m = − n . Finally, since any group BS ( m,n ) can be viewed as a HNN-extension, then we point out the Ribes-Zalesskii property of rank two on some HNN-extensions.