Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-24T01:40:19.280Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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