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Shift dynamics of the groups of Fibonacci type

Shift dynamics of the groups of Fibonacci type AbstractWe study the shift dynamics of the groups G=Gn⁢(x0⁢xm⁢xk-1)G=G_{n}(x_{0}x_{m}x_{k}^{-1})of Fibonacci type introduced by Johnson and Mawdesley.The main result concerns the order of the shift automorphism of 𝐺 and determining whether it is an outer automorphism, and we find the latter occurs if and only if 𝐺 is not perfect.A result of Bogley provides that the aspherical presentations determine groups admitting a free shift action by Zn\mathbb{Z}_{n}on the nonidentity elements of 𝐺, from which it follows that the shift is an outer automorphism of order 𝑛 when 𝐺 is nontrivial.The focus of this paper is therefore on the non-aspherical cases, which include for example the Fibonacci and Sieradski groups.With few exceptions, the fixed-point and freeness problems for the shift automorphism are solved, in some cases using computational and topological methods. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Group Theory de Gruyter

Shift dynamics of the groups of Fibonacci type

Journal of Group Theory , Volume 26 (1): 28 – Jan 1, 2023

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References (18)

Publisher
de Gruyter
Copyright
© 2022 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-4446
eISSN
1435-4446
DOI
10.1515/jgth-2022-0003
Publisher site
See Article on Publisher Site

Abstract

AbstractWe study the shift dynamics of the groups G=Gn⁢(x0⁢xm⁢xk-1)G=G_{n}(x_{0}x_{m}x_{k}^{-1})of Fibonacci type introduced by Johnson and Mawdesley.The main result concerns the order of the shift automorphism of 𝐺 and determining whether it is an outer automorphism, and we find the latter occurs if and only if 𝐺 is not perfect.A result of Bogley provides that the aspherical presentations determine groups admitting a free shift action by Zn\mathbb{Z}_{n}on the nonidentity elements of 𝐺, from which it follows that the shift is an outer automorphism of order 𝑛 when 𝐺 is nontrivial.The focus of this paper is therefore on the non-aspherical cases, which include for example the Fibonacci and Sieradski groups.With few exceptions, the fixed-point and freeness problems for the shift automorphism are solved, in some cases using computational and topological methods.

Journal

Journal of Group Theoryde Gruyter

Published: Jan 1, 2023

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