Distortion element in the automorphism group of a full shift
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2022
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| _version_ | 1866917840870703104 |
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| author | Callard, Antonin Salo, Ville |
| author_facet | Callard, Antonin Salo, Ville |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_00685 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Distortion element in the automorphism group of a full shift Callard, Antonin Salo, Ville Group Theory Computational Complexity Dynamical Systems 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. |
| title | Distortion element in the automorphism group of a full shift |
| topic | Group Theory Computational Complexity Dynamical Systems |
| url | https://arxiv.org/abs/2208.00685 |