Shift dynamics of the groups of Fibonacci type

Abstract We study the shift dynamics of the groups G = G n ⁢ ( x 0 ⁢ x m ⁢ x k - 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 Z n \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.