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Distortion element in the automorphism group of a full shift
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 7 / July 2024
- Published online by Cambridge University Press:
- 23 October 2023, pp. 1757-1817
- Print publication:
- July 2024
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- Article
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We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G, and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups
$mV$ admit distortion elements; in particular, 
$2V$ (unlike V) does not admit a proper action on a CAT
$(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci. 84 (2017), 288–301].
