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Commutator length of powers in free products of groups

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

Published online by Cambridge University Press:  27 December 2021

Vadim Yu. Bereznyuk  and
Anton A. Klyachko
Vadim Yu. Bereznyuk
Affiliation:
Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow119991, Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia ([email protected]; [email protected])
Anton A. Klyachko
Affiliation:
Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow119991, Russia Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia ([email protected]; [email protected])
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Abstract

Given groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.

Keywords

commutator lengthstable commutator lengthfree products of groups

MSC classification

Primary: 57M07: Topological methods in group theory
Secondary: 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F06: Cancellation theory; application of van Kampen diagrams 20F12: Commutator calculus 20F70: Algebraic geometry over groups; equations over groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 1 , February 2022 , pp. 102 - 119
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Calegari, D., scl. MSJ Memoirs, 20, pp. xii+209 (Mathematical Society of Japan, Tokyo, 2009).Google Scholar
Chen, L., Spectral gap of scl in free products, Proc. Amer. Math. Soc. 146(7) (2018), 3143–3151. arXiv:1611.07936.CrossRefGoogle Scholar
Clifford, A., Non-amenable type K equations over groups, Glasgow Math. J. 45(2) (2003), 389–400.CrossRefGoogle Scholar
Clifford, A. and Goldstein, R. Z., Tesselations of $S^{2}$ and equations over torsion-free groups, Proc. Edinburgh Math. Soc. 38(3) (1995), 485–493.CrossRefGoogle Scholar
Clifford, A. and Goldstein, R. Z., The group $\langle G,\,t|e\rangle$ when $G$ is torsion free, J. Algebra 245(1) (2001), 297–309.CrossRefGoogle Scholar
Cohen, M. M. and Rourke, C., The surjectivity problem for one-generator, one-relator extensions of torsion-free groups, Geom. Topology 5(1) (2001), 127–142. arXiv:math/0009101CrossRefGoogle Scholar
Comerford, J. A., Comerford, L. P. Jr. and Edmunds, C. C., Powers as products of commutators, Comm. Algebra 19(2) (1991), 675–684.CrossRefGoogle Scholar
Comerford, L. P. Jr., Edmunds, C. C. and Rosenberger, G., Commutators as powers in free products of groups, Proc. Amer. Math. Soc. 122(1) (1994), 47–52. arXiv:math/9310205CrossRefGoogle Scholar
Culler, M., Using surfaces to solve equations in free groups, Topology 20(2) (1981), 133–145.CrossRefGoogle Scholar
Duncan, A. J. and Howie, J., The genus problem for one-relator products of locally indicable groups, Math. Z. 208(1) (1991), 225–237.CrossRefGoogle Scholar
Fenn, R. and Rourke, C., Klyachko's methods and the solution of equations over torsion-free groups, L'Enseignement Math. 42 (1996), 49–74.Google Scholar
Forester, M. and Rourke, C., Diagrams and the second homotopy group, Commun. Anal. Geom. 13(4) (2005), 801–820. arXiv:math/0306088CrossRefGoogle Scholar
Forester, M. and Rourke, C., The adjunction problem over torsion-free groups, Proc. Natl. Acad. Sci. USA 102(36) (2005), 12670–12671. arXiv:math/0412274CrossRefGoogle ScholarPubMed
Frenkel, E. V. and Klyachko, Ant. A., Commutators cannot be proper powers in metric small-cancellation torsion-free groups, arXiv:1210.7908Google Scholar
Giang, Le Thi, The relative hyperbolicity of one-relator relative presentations, J. Group Theory 12(6) (2009), 949–959. arXiv:0807.2487CrossRefGoogle Scholar
Howie, J., The solution of length three equations over groups, Proc. Edinburgh Math. Soc. 26(1) (1983), 89–96.CrossRefGoogle Scholar
Howie, J., The quotient of a free product of groups by a single high-powered relator. II. Fourth powers, Proc. London Math. Soc. s3-61(1) (1990), 33–62.CrossRefGoogle Scholar
Ivanov, S. V. and Klyachko, Ant. A., Solving equations of length at most six over torsion-free groups, J. Group Theory 3(3) (2000), 329–337.CrossRefGoogle Scholar
Ivanov, S. V. and Klyachko, Ant. A., Quasiperiodic and mixed commutator factorizations in free products of groups, Bull. London Math. Soc. 50(5) (2018), 832–844. arXiv:1702.01379CrossRefGoogle Scholar
Klyachko, Ant. A., A funny property of a sphere and equations over groups, Comm. Algebra 21(7) (1993), 2555–2575.CrossRefGoogle Scholar
Klyachko, Ant. A., Asphericity tests, Internat. J. Algebra Comp. 7(4) (1997), 415–431.CrossRefGoogle Scholar
Klyachko, Ant. A., The Kervaire–Laudenbach conjecture and presentations of simple groups, Algebra Logic 44(4) (2005), 219–242. arXiv:math.GR/0409146CrossRefGoogle Scholar
Klyachko, Ant. A., How to generalize known results on equations over groups, Math. Notes 79(3–4) (2006), 377–386. arXiv:math.GR/0406382CrossRefGoogle Scholar
Klyachko, Ant. A., SQ-universality of one-relator relative presentations, Sbornik: Math. 197(10) (2006), 1489–1508. arXiv:math.GR/0603468CrossRefGoogle Scholar
Klyachko, Ant. A., Free subgroups of one-relator relative presentations, Algebra Logic 46(3) (2007), 158–162. arXiv:math.GR/0510582CrossRefGoogle Scholar
Klyachko, Ant. A., The structure of one-relator relative presentations and their centres, J. Group Theory 12(6) (2009), 923–947. arXiv:math.GR/0701308CrossRefGoogle Scholar
Klyachko, Ant. A. and Lurye, D. E., Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator, J. Pure Appl. Algebra 216(3) (2012), 524–534. arXiv:1010.4220CrossRefGoogle Scholar
Mertens, S., The easiest hard problem: number partitioning, Comput. Complex. Stat. Phys. 125(2) (2006), 125–139. arXiv:cond-mat/0310317Google Scholar
Schützenberger, M. P., Sur l'equation $a^{2+n}=b^{2}+mc^{2}+p$ dans un groupe libre, C. R. Acad. Sci. Paris Sér. I Math. 248 (1959), 2435–2436.Google Scholar