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Commutator length of powers in free products of groups

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

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 (kuynzereb@gmail.com; klyachko@mech.math.msu.su)
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 (kuynzereb@gmail.com; klyachko@mech.math.msu.su)
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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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