| Title:
|
The Ribes-Zalesskii property of some one relator groups (English) |
| Author:
|
Mantika, Gilbert |
| Author:
|
Temate-Tangang, Narcisse |
| Author:
|
Tieudjo, Daniel |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
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58 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
35-47 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
profinite topology |
| Keyword:
|
HNN-extension |
| Keyword:
|
Ribes-Zalesskii property of rank $k$ |
| Keyword:
|
Baumslag-Solitar groups |
| MSC:
|
20E06 |
| MSC:
|
20E26 |
| MSC:
|
20F05 |
| MSC:
|
22A05 |
| idZBL:
|
Zbl 07511506 |
| idMR:
|
MR4412965 |
| DOI:
|
10.5817/AM2022-1-35 |
| . |
| Date available:
|
2022-02-23T12:11:14Z |
| Last updated:
|
2022-06-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149445 |
| . |
| Reference:
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