Distortion element in the automorphism group of a full shift

08/01/2022

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by   Antonin Callard, et al.

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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. The distortion element is essentially the SMART machine.