Distortion element in the automorphism group of a full shift

TL;DR

This paper demonstrates the existence of distortion elements in automorphism groups of full shifts, revealing new structural properties and implications for subshifts, sofic shifts, and certain groups like Turing machines and Brin-Thompson groups.

Contribution

It introduces the first known distortion element in the automorphism group of a full shift and explores its implications for various classes of subshifts and groups.

Findings

Existence of a distortion element in the automorphism group of a full shift.

Lower bounds on entropy dimension for subshifts containing such groups.

Characterization of distortion elements in sofic shifts and specific groups.

Abstract

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.