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The Klein bottle group is not strongly verbally closed, though awfully close to being so

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  03 August 2020

Anton A. Klyachko
Anton A. Klyachko*
Affiliation:
Faculty of Mechanics and Mathematics of Moscow State University, Moscow, Leninskie gory, MSU, Russia, 119991
*
e-mail: [email protected]
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Abstract

According to Mazhuga’s theorem, the fundamental group H of anyconnected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing H as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.

Keywords

Verbally closed subgroupssurface groups

MSC classification

Primary: 20F70: Algebraic geometry over groups; equations over groups 20F34: Fundamental groups and their automorphisms 20F16: Solvable groups, supersolvable groups
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 491 - 497
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

This work was supported by the Russian Foundation for Basic Research, project no. 19-01-00591.

References

Baumslag, G., Myasnikov, A., and Remeslennikov, V., Algebraic geometry over groups I. Algebraic sets and ideal theory. J. Algebra 219(1999), 16–79. https://doi.org/10.1006/jabr.1999.7881 CrossRefGoogle Scholar
Bogopolski, O., Equations in acylindrically hyperbolic groups and verbal closedness. Preprint, 2020. arXiv:1805.08071 Google Scholar
Bogopolski, O., On finite systems of equations in acylindrically hyperbolic groups. Preprint, 2020. arXiv:1903.10906 CrossRefGoogle Scholar
Klyachko, A. A. and Mazhuga, A. M., Verbally closed virtually free subgroups. Sbornik: Mathematics 209(2018), 850–856. http://dx.doi.org/10.1070/SM8942 CrossRefGoogle Scholar
Klyachko, A. A., Mazhuga, A. M., and Miroshnichenko, V. Yu., Virtually free finite-normal-subgroup-free groups are strongly verbally closed. J. Algebra 510(2018), 319–330. https://doi.org/10.1016/j.jalgebra.2018.05.028CrossRefGoogle Scholar
Lee, D., On certain C-test words for free groups. J. Algebra 247(2002), 509–540. https://doi.org/10.1006/jabr.2001.9001 CrossRefGoogle Scholar
Mazhuga, A. M., On free decompositions of verbally closed subgroups of free products of finite groups. J. Group Theory 20(2016). https://doi.org/10.1515/jgth-2016-0058 Google Scholar
Mazhuga, A. M., Strongly verbally closed groups. J. Algebra 493(2018), 171–184. https://doi.org/10.1016/j.jalgebra.2017.09.021 CrossRefGoogle Scholar
Mazhuga, A. M., Free products of groups are strongly verbally closed . Sbornik: Mathematics 210(2019), 1456–1492. http://dx.doi.org/10.1070/SM9115 CrossRefGoogle Scholar
Myasnikov, A. and Roman’kov, V., Verbally closed subgroups of free groups . J. Group Theory 17(2014), 29–40. https://doi.org/10.1515/jgt-2013-0034 CrossRefGoogle Scholar
Newman, M., A note on Fuchsian groups . Illinois J. Math. 29(1985), 682–686. https://dx.doi.org/10.1215/ijm/1256045503 CrossRefGoogle Scholar
Roman’kov, V. A., Equations over groups . Groups Complexity Cryptology 4(2012), 191–239. https://doi.org/10.1515/gcc-2012-0015 Google Scholar
Roman’kov, V. A. and Khisamiev, N. G., Verbally and existentially closed subgroups of free nilpotent groups. Algebra Logic 52(2013), 336–351. https://doi.org/10.1007/S10469-013-9245-6 CrossRefGoogle Scholar
Roman’kov, V. A. and Timoshenko, E. I., On verbally closed subgroups of free solvable groups . Vestnik Omskogo Universiteta 24(2019), 9–16 [in Russian]. https://doi.org/10.25513/1812-3996.2019.24(1).9-16 Google Scholar