Distortion element in the automorphism group of a full shift

Autor:	Callard, Antonin, Salo, Ville
Rok vydání:	2022
Předmět:	Mathematics - Group Theory Computer Science - Computational Complexity Mathematics - Dynamical Systems
Druh dokumentu:	Working Paper
Popis:	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. Comment: 64 pages, 6 figures; improved exposition
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2208.00685 Zobrazit plný text záznamu View this record from Arxiv