دورية أكاديمية
The Ribes-Zalesskii property of some one relator groups
| العنوان: | The Ribes-Zalesskii property of some one relator groups |
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| المؤلفون: | Mantika, Gilbert, Temate-Tangang, Narcisse, Tieudjo, Daniel |
| بيانات النشر: | Department of Mathematics, Faculty of Science of Masaryk University, Brno |
| سنة النشر: | 2022 |
| المجموعة: | DML-CZ (Czech Digital Mathematics Library) |
| مصطلحات موضوعية: | 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 |
| الوصف: | 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. |
| نوع الوثيقة: | text |
| وصف الملف: | application/pdf |
| اللغة: | English |
| تدمد: | 0044-8753 1212-5059 |
| العلاقة: | mr:MR4412965; zbl:Zbl 07511506; reference:[1] Allenby, R., Doniz, D.: A free product of finitely generated nilpotent groups amalgamating a cycle that is not subgroup separable.Proc. Amer. Math. Soc. 124 (4) (1996), 1003–1005. 10.1090/S0002-9939-96-03567-8; reference:[2] Baumslag, B., Tretkoff, M.: Residually finite HNN extensions.Comm. Algebra 6 (1978), 179–194. 10.1080/00927877808822240; reference:[3] Baumslag, G.: On the residual finiteness of generalised free products of nilpotent groups.Trans. Amer. Math. Soc. 106 (1963), 193–209. 10.1090/S0002-9947-1963-0144949-8; reference:[4] Baumslag, G., Solitar, D.: Some two-generator one-relator non-Hopfien groups.Bull. Amer. Math. Soc. 38 (1962), 199–201.; reference:[5] Cohen, D.: Combinatorial groups theory: a topological approach.London Math. Soc. Stud. Texts, 1989, ISBN: 0-521-34133-7; 0-521-34936-2.; reference:[6] Coulbois, T.: Propriètés de Ribes-Zalesskii, topologie profinie, produit libre et généralisations.Ph.D. thesis, Universite Paris vii-Denis Diderot, 2000.; reference:[7] Coulbois, T.: Free products, profinite topology and finitely generated subgroups.Internat. J. Algebra Comput. 14 (2001), 171–184. MR 1829049, 10.1142/S0218196701000449; reference:[8] Doucha, M., Malicki, M.: Generic representations of countable groups.Trans. Amer. Math. Soc. 164 (2019), 105–114. MR 4029696; reference:[9] Hall, M.: Cosets representations in free groups.Trans. Amer. Math. Soc. 67 (1949), 421–432. 10.1090/S0002-9947-1949-0032642-4; reference:[10] Hall, M.: A topology for free groups and related groups.Ann. of Math. 52 (1950), 127–139. 10.2307/1969513; reference:[11] Hall, P.: On the finiteness on certain soluble groups.Proc. London Math. Soc. 164 (1959), 595–622.; reference:[12] Hirsch, K.: On infinite soluble groups. III.Proc. London Math. Soc. 49 (1946), 184–194. 10.1112/plms/s2-49.3.184; reference:[13] Lennox, J.C., Wilson, J.S.: On products of subgroups in polycyclic groups.Arch. Math. (Basel) 33 (1979), 305–309. 10.1007/BF01222760; reference:[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.; reference:[15] Mal’cev, A.: On homomorphisms onto finite groups.Ivanov. Gos. Ped. Inst. Ucen. Zap. 18 (1958), 49–60.; reference:[16] Meskin, S.: Nonresidually finite one relator groups.Trans. Amer. Math. Soc. 164 (1972), 105–114. 10.1090/S0002-9947-1972-0285589-5; reference:[17] Metaftsis, V., Raptis, E.: Subgroups Separability of HNN-Extensions with Abelian Base Groups.J. Algebra 245 (2001), 42–49. MR 1868182, 10.1006/jabr.2001.8910; reference:[18] Moldavanskii, D., Uskova, A.: On the finitely separability of subgroups of generalized free products.Arxiv: 1308.3955.[mathGR], (2013).; reference:[19] Neumann, B.H.: An essay of free products of groups with amalgamations.Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. 10.1098/rsta.1954.0007; reference:[20] Pin, J.-E., Reutenaeur, C.: A conjecture on the Hall topology for free groups.Bull. London Math. Soc. 23 (1991), 356–362. 10.1112/blms/23.4.356; reference:[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. MR 3677322; reference:[22] Ribes, L., Zalesskii, P.A.: Profinite topologies in free products of group.Internat. J. Algebra Comput. 14 (2004), 751–772. MR 2104779, 10.1142/S0218196704001992; reference:[23] Rips, E.: An example of a NON-LERF group which is a free product of LERF groups with an amalgamated cyclic subgroup.Israel J. Math. Vol. 70 (1) (1990), 104–110. 10.1007/BF02807222; reference:[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.; reference:[25] Rosendal, C.: Finitely approximable groups and actions Part I: The Ribes-Zalesskii property.J. Symbolic Logic 76 (4) (2011), 1297–1306. MR 2895386, 10.2178/jsl/1318338850; reference:[26] Rosendal, C.: Finitely approximable groups and actions Part II: Generic representations.J. Symbolic Logic 76 (4) (2011), 1307–1321. MR 2895397, 10.2178/jsl/1318338851; reference:[27] Shihong, You: The product separability of the generalized free products of cyclic groups.J. London Math. Soc.(2) 56 (1997), 91–103. 10.1112/S0024610796004425; reference:[28] Stebe, P.: Conjugacy separability of certain free products with amalgamation.Trans. Amer. Math. Soc. 156 (1971), 119–129. 10.1090/S0002-9947-1971-0274597-5 |
| الإتاحة: | https://doi.org/10.5817/AM2022-1-35Test http://hdl.handle.net/10338.dmlcz/149445Test |
| حقوق: | access:Unrestricted ; rights:DML-CZ Czech Digital Mathematics Library, http://dml.czTest/ ; rights:Institute of Mathematics AS CR, http://www.math.cas.czTest/ ; conditionOfUse:http://dml.cz/useTest |
| رقم الانضمام: | edsbas.B9943C0D |
| قاعدة البيانات: | BASE |
Full Text Finder | تدمد: | 00448753 12125059 |
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