Address:
Department of Mathematics, Higher Teacher’s Training College, The University of Maroua, P.O. Box 55 Maroua – Cameroon
Department of Mathematics and Computer Science, Faculty of Sciences, The University of Maroua, P.O. Box 814 Maroua – Cameroon
Department of Mathematics and Computer Science, The University of Ngaoundere and AIMS Cameroon, P.O. Box 454 Ngaoundere – Cameroon
E-mail:
gilbertmantika@yahoo.fr
tematen@yahoo.fr
tieudjo@yahoo.com
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 $H_{1} H_{2} \cdots H_{k}$ of finitely generated subgroups $H_{1}, H_{2}, \cdots , H_{k}$ 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.
AMSclassification: primary 20E06; secondary 20E26, 20F05, 22A05.
Keywords: profinite topology, HNN-extension, Ribes-Zalesskii property of rank k, Baumslag-Solitar groups.
DOI: 10.5817/AM2022-1-35