Distortion element in the automorphism group of a full shift

Abstract

<p>We show that there is a distortion element in a finitely generated subgroup <em>G</em> 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 <em>G</em>, 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 <em>mV</em> admit distortion elements; in particular, 2<em>V</em> (unlike <em>V</em>) 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. <em>J. Comput. System Sci.</em> <strong>84</strong> (2017), 288–301].</p>