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].

中文翻译：

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全平移自同构群中的畸变元

我们证明在有限生成的子群中存在畸变元素G全移的自同构群，即词范数呈多对数增长的无限阶元素。作为推论，我们获得了包含以下副本的任何子移位的熵维度的下界G，并且当且仅当 sofic 移位是不可数时，sofic 移位的自同构群包含畸变元素。我们还获得了图灵机群和高维布林-汤普森群 $mV$ 允许畸变元素；尤其， $2V$ （不像V) 不承认对 CAT 采取适当的行动 $(0)$ 立方体复合体。在每种情况下，畸变元件大致对应于 Cassaigne、Ollinger 和 Torres-Avilés 的 SMART 机 [一种小型最小非周期可逆图灵机。J. 计算机。系统科学。84（2017），288-301]。