The Ribes-Zalesskii property of some one relator groups

Gilbert Mantika; Narcisse Temate-Tangang; Daniel Tieudjo

Archivum Mathematicum (2022)

  • Volume: 058, Issue: 1, page 35-47
  • ISSN: 0044-8753

Abstract

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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 B S ( 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 B S ( m , n ) can be viewed as a HNN-extension, then we point out the Ribes-Zalesskii property of rank two on some HNN-extensions.

How to cite

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Mantika, Gilbert, Temate-Tangang, Narcisse, and Tieudjo, Daniel. "The Ribes-Zalesskii property of some one relator groups." Archivum Mathematicum 058.1 (2022): 35-47. <http://eudml.org/doc/298251>.

@article{Mantika2022,
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.},
author = {Mantika, Gilbert, Temate-Tangang, Narcisse, Tieudjo, Daniel},
journal = {Archivum Mathematicum},
keywords = {profinite topology; HNN-extension; Ribes-Zalesskii property of rank $k$; Baumslag-Solitar groups},
language = {eng},
number = {1},
pages = {35-47},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {The Ribes-Zalesskii property of some one relator groups},
url = {http://eudml.org/doc/298251},
volume = {058},
year = {2022},
}

TY - JOUR
AU - Mantika, Gilbert
AU - Temate-Tangang, Narcisse
AU - Tieudjo, Daniel
TI - The Ribes-Zalesskii property of some one relator groups
JO - Archivum Mathematicum
PY - 2022
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 058
IS - 1
SP - 35
EP - 47
AB - 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.
LA - eng
KW - profinite topology; HNN-extension; Ribes-Zalesskii property of rank $k$; Baumslag-Solitar groups
UR - http://eudml.org/doc/298251
ER -

References

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  1. Allenby, R., Doniz, D., 10.1090/S0002-9939-96-03567-8, Proc. Amer. Math. Soc. 124 (4) (1996), 1003–1005. (1996) DOI10.1090/S0002-9939-96-03567-8
  2. Baumslag, B., Tretkoff, M., 10.1080/00927877808822240, Comm. Algebra 6 (1978), 179–194. (1978) DOI10.1080/00927877808822240
  3. Baumslag, G., 10.1090/S0002-9947-1963-0144949-8, Trans. Amer. Math. Soc. 106 (1963), 193–209. (1963) DOI10.1090/S0002-9947-1963-0144949-8
  4. Baumslag, G., Solitar, D., Some two-generator one-relator non-Hopfien groups, Bull. Amer. Math. Soc. 38 (1962), 199–201. (1962) 
  5. Cohen, D., Combinatorial groups theory: a topological approach, London Math. Soc. Stud. Texts, 1989, ISBN: 0-521-34133-7; 0-521-34936-2. (1989) 
  6. Coulbois, T., Propriètés de Ribes-Zalesskii, topologie profinie, produit libre et généralisations, Ph.D. thesis, Universite Paris vii-Denis Diderot, 2000. (2000) 
  7. Coulbois, T., 10.1142/S0218196701000449, Internat. J. Algebra Comput. 14 (2001), 171–184. (2001) MR1829049DOI10.1142/S0218196701000449
  8. Doucha, M., Malicki, M., Generic representations of countable groups, Trans. Amer. Math. Soc. 164 (2019), 105–114. (2019) MR4029696
  9. Hall, M., 10.1090/S0002-9947-1949-0032642-4, Trans. Amer. Math. Soc. 67 (1949), 421–432. (1949) DOI10.1090/S0002-9947-1949-0032642-4
  10. Hall, M., 10.2307/1969513, Ann. of Math. 52 (1950), 127–139. (1950) DOI10.2307/1969513
  11. Hall, P., On the finiteness on certain soluble groups, Proc. London Math. Soc. 164 (1959), 595–622. (1959) 
  12. Hirsch, K., 10.1112/plms/s2-49.3.184, Proc. London Math. Soc. 49 (1946), 184–194. (1946) DOI10.1112/plms/s2-49.3.184
  13. Lennox, J.C., Wilson, J.S., 10.1007/BF01222760, Arch. Math. (Basel) 33 (1979), 305–309. (1979) DOI10.1007/BF01222760
  14. Magnus, W., Karass, A., Solitar, D., Combinatorial Group Theory: Presentations of Groups in Term of Generators and Relations, Interscience Publishers (John Wiley and Sons), New York, 1966. (1966) 
  15. Mal’cev, A., On homomorphisms onto finite groups, Ivanov. Gos. Ped. Inst. Ucen. Zap. 18 (1958), 49–60. (1958) 
  16. Meskin, S., 10.1090/S0002-9947-1972-0285589-5, Trans. Amer. Math. Soc. 164 (1972), 105–114. (1972) DOI10.1090/S0002-9947-1972-0285589-5
  17. Metaftsis, V., Raptis, E., 10.1006/jabr.2001.8910, J. Algebra 245 (2001), 42–49. (2001) MR1868182DOI10.1006/jabr.2001.8910
  18. Moldavanskii, D., Uskova, A., On the finitely separability of subgroups of generalized free products, Arxiv: 1308.3955.[mathGR], (2013). 
  19. Neumann, B.H., 10.1098/rsta.1954.0007, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. (1954) DOI10.1098/rsta.1954.0007
  20. Pin, J.-E., Reutenaeur, C., 10.1112/blms/23.4.356, Bull. London Math. Soc. 23 (1991), 356–362. (1991) DOI10.1112/blms/23.4.356
  21. Ribes, L., Profinite graphs and groups, Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, Springer, Cham, 2017, ISBN: 978-3-319-61041-2; 978-3-319-61199-0. (2017) MR3677322
  22. Ribes, L., Zalesskii, P.A., 10.1142/S0218196704001992, Internat. J. Algebra Comput. 14 (2004), 751–772. (2004) MR2104779DOI10.1142/S0218196704001992
  23. Rips, E., 10.1007/BF02807222, Israel J. Math. Vol. 70 (1) (1990), 104–110. (1990) DOI10.1007/BF02807222
  24. Romanovskii, N.S., On the residual finiteness of free products with respect to subgroups, Izv. Akad. Nauk SSSR Ser. Math. 33 (1969), 1324–1329. (1969) 
  25. Rosendal, C., 10.2178/jsl/1318338850, J. Symbolic Logic 76 (4) (2011), 1297–1306. (2011) MR2895386DOI10.2178/jsl/1318338850
  26. Rosendal, C., 10.2178/jsl/1318338851, J. Symbolic Logic 76 (4) (2011), 1307–1321. (2011) MR2895397DOI10.2178/jsl/1318338851
  27. Shihong, You, 10.1112/S0024610796004425, J. London Math. Soc.(2) 56 (1997), 91–103. (1997) DOI10.1112/S0024610796004425
  28. Stebe, P., 10.1090/S0002-9947-1971-0274597-5, Trans. Amer. Math. Soc. 156 (1971), 119–129. (1971) DOI10.1090/S0002-9947-1971-0274597-5

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