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.

翻译：我们证明全转移的自同构群的一个有限生成子群 $G$ 中存在扭曲元素，即一个具有无限阶且其词范数呈多对数增长的群元素。作为推论，我们得到了包含 $G$ 的任意子移的熵维数的下界，并证明一个sofic子移的自同构群包含扭曲元素当且仅当该sofic子移是不可数的。我们还得到图灵机群以及高维Brin-Thompson群 $mV$ 均包含扭曲元素；特别地，$2V$（与 $V$ 不同）在CAT$(0)$立方复形上没有真作用。该扭曲元素本质上是SMART机。