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Yet another Freiheitssatz: Mating finite groups with locally indicable ones

Part of: Low-dimensional topology Special aspects of infinite or finite groups

Published online by Cambridge University Press:  14 November 2022

Anton A. Klyachko  and
Mikhail A. Mikheenko
Anton A. Klyachko*
Affiliation:
Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, MSU, Moscow 119991, Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia
Mikhail A. Mikheenko
Affiliation:
Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, MSU, Moscow 119991, Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia
*
*Corresponding author. E-mail: klyachko@mech.math.msu.su
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Abstract

The main result includes as special cases on the one hand, the Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii–Howie–Short theorem (1980–1984) generalising Magnus’s Freiheitssatz (1930).

Keywords

equations over groupsFreiheitssatz

MSC classification

Primary: 20F70: Algebraic geometry over groups; equations over groups
Secondary: 20F65: Geometric group theory 57M07: Topological methods in group theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 65 , Issue 2 , May 2023 , pp. 337 - 344
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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Footnotes

*

Definitions, examples, and properties of hyperlinear (= Connes-embeddable) groups can be found, e.g., in [36]; we note only that the class of hyperlinear groups contains all finite group and their free products (possibly, even all group are hyperlinear – this is a well-known open question).

This work was supported by the Russian Science Foundation, project no. 22-11-00075.

References

Anwar, M. F., Bibi, M., and Akram, M. S., On solvability of certain equations of arbitrary length over torsion-free groups, Glasg. Math. J., 63(3) (2021), 651–659.10.1017/S0017089520000427CrossRefGoogle Scholar
Baranov, D. V. and Klyachko, A. A., Economical adjunction of square roots to groups, Siberian Math. J., 53(2) (2012), 201–206.10.1134/S0037446612020024CrossRefGoogle Scholar
Brodskii, S. D., Equations over groups and group with a single defining relation, Russian Math. Surv., 35(4) (1980), 165–165.10.1070/RM1980v035n04ABEH001865CrossRefGoogle Scholar
Brodskii, S. D., Equations over groups and group with one relator, Siberian Math. J., 25(2) (1984), 235–251.10.1007/BF00971461CrossRefGoogle Scholar
Brown, N., Dykema, K. and Jung, K., Free entropy dimension in amalgamated free products, Proc. London Math. Soc., 97(2) (2008), 339–367.10.1112/plms/pdm054CrossRefGoogle Scholar
Bibi, M. and Edjvet, M., Solving equations of length seven over torsion-free groups, J. Group Theory, 21(1) (2018), 147–164.10.1515/jgth-2017-0032CrossRefGoogle Scholar
Clifford, A. and Goldstein, R. Z., Equations with torsion-free coefficients, Proc. Edinburgh Math. Soc., 43(2) (2000), 295–307.10.1017/S0013091500020939CrossRefGoogle Scholar
Edjvet, M. and Howie, J., The solution of length four equations over groups, Trans. Amer. Math. Soc., 326(1) (1991), 345–369.10.1090/S0002-9947-1991-1002920-5CrossRefGoogle Scholar
Edjvet, M. and Howie, J., On singular equations over torsion-free groups, International Journal of Algebra and Computation, 31:3 (2021), 551–580.10.1142/S0218196721500272CrossRefGoogle Scholar
Edjvet, M. and Juhász, A., Equations of length 4 and one-relator products, Math. Proc. Cambridge Phil. Soc., 129(2) (2000), 217–230.10.1017/S0305004100004539CrossRefGoogle Scholar
Fenn, R. and Rourke, C., Klyachko’s methods and the solution of equations over torsion-free groups, L’Enseignment Mathématique, 42 (1996), 49–74.Google Scholar
Gerstenhaber, M. and Rothaus, O. S., The solution of sets of equations in groups, Proc. Nat. Acad. Sci. USA, 48(9) (1962), 1531–1533.10.1073/pnas.48.9.1531CrossRefGoogle ScholarPubMed
Gruenberg, K. W., Residual properties of infinite soluble groups, Proc. London Math. Soc., s3–7(1) (1957), 29–62.10.1112/plms/s3-7.1.29CrossRefGoogle Scholar
Higman, G., The units of group-rings, Proc. London Math. Soc., s2–46(1) (1940), 231–248.10.1112/plms/s2-46.1.231CrossRefGoogle Scholar
Howie, J., On pairs of 2-complexes and systems of equations over groups, J. Reine Angew Math., 1981(324) (1981), 165–174.Google Scholar
Howie, J., The quotient of a free product of groups by a single high-powered relator. III: The word problem, Proc. Lond. Math. Soc., 62(3) (1991), 590–606.10.1112/plms/s3-62.3.590CrossRefGoogle Scholar
Ivanov, S. V. and Klyachko, A. A., Solving equations of length at most six over torsion-free groups, J. Group Theory, 3(3) (2000), 329–337.10.1515/jgth.2000.026CrossRefGoogle Scholar
Juhászasz, A., On the solvability of a class of equations over groups, Math. Proc. Cambridge Phil. Soc., 135(2) (2003), 211–217.10.1017/S0305004103006765CrossRefGoogle Scholar
Kargapolov, M. I. and Merzljakov, J. I., Fundamentals of the theory of groups. Graduate Texts in Mathematics, vol. 62 (Springer, 1979).10.1007/978-1-4612-9964-6CrossRefGoogle Scholar
Klyachko, A. A., A funny property of sphere and equations over groups, Commun. Algebra, 21(7) (1993), 2555–2575.10.1080/00927879308824692CrossRefGoogle Scholar
Klyachko, A. A., Asphericity tests, Int. J. Algebra Comput., 7(4) (1997), 415–431.10.1142/S0218196797000186CrossRefGoogle Scholar
Klyachko, A. A., Equations over groups, quasivarieties, and a residual property of a free group, Journal of Group Theory, 2(3) (1999), 319–327.10.1515/jgth.1999.021CrossRefGoogle Scholar
Klyachko, A. A., How to generalize known results on equations over groups, Math. Notes, 79(3) (2006), 377–386.Google Scholar
Klyachko, A. A. and Prishchepov, M. I., The descent method for equations over groups, Moscow Univ. Math. Bull. 50(4) (1995), 56–58.Google Scholar
Klyachko, A. A. and Thom, A. B., New topological methods to solve equations over groups, Algebraic Geom. Topol., 17(1) (2017), 331–353.10.2140/agt.2017.17.331CrossRefGoogle Scholar
Krasner, M. and Kaloujnine, L., Produit complet des groupes de permutations et le problème d’extension de groupes. III, Acta Sci. Math., 14 (1951), 69–82.Google Scholar
Levin, F., Solutions of equations over groups, Bull. Amer. Math. Soc., 68(6) (1962), 603–604.10.1090/S0002-9904-1962-10868-4CrossRefGoogle Scholar
Lyndon, R. C., Equations in groups, Bol. Soc. Bras. Math., 11(1) (1980), 79–102.10.1007/BF02584882CrossRefGoogle Scholar
Lyndon, R. and Schupp, P., Combinatorial group theory (Springer, 1977).Google Scholar
Magnus, W., Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz), J. Reine Angew Math., 163 (1930), 141–165.10.1515/crll.1930.163.141CrossRefGoogle Scholar
Mal’cev, A. I., Algebraic systems. Springer, 1973.10.1007/978-3-642-65374-2CrossRefGoogle Scholar
Nitsche, M. and Thom, A., Universal solvability of group equations, J. Group Theory, 25(1) (2022), 1–10.CrossRefGoogle Scholar
Pestov, V. G., Hyperlinear and sofic groups: A brief guide, Bull. Symb. Log., 14(4) (2008), 449–480.10.2178/bsl/1231081461CrossRefGoogle Scholar
Schreier, O., Die Untergruppen der freien Gruppen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5(1) (1927), 161–183.10.1007/BF02952517CrossRefGoogle Scholar
Short, H., Topological methods in group theory: the adjunction problem. Ph.D. Thesis (University of Warwick, 1981).Google Scholar
Thom, A., Finitary approximations of groups and their applications, in Proceedings of the International Congress of Mathematicians — Rio de Janeiro 2018. Vol. III. Invited lectures, World Scientific, Hackensack (2018), 1779–1799.10.1142/9789813272880_0117CrossRefGoogle Scholar
Zieschang, H., Vogt, E. and Coldewey, H.-D., Surfaces and planar discontinuous groups. (Springer, 1980).10.1007/BFb0089692CrossRefGoogle Scholar