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AbstractWe study the shift dynamics of the groups G=Gn(x0xmxk-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 of Group Theory – de Gruyter
Published: Jan 1, 2023
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