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Distortion element in the automorphism group of a full shift

Antonin Callard, Ville Salo

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.

Distortion element in the automorphism group of a full shift

Abstract

We show that there is a distortion element in a finitely-generated subgroup 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 , 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 admit distortion elements; in particular, (unlike ) does not admit a proper action on a CAT cube complex. The distortion element is essentially the SMART machine.
Paper Structure (50 sections, 40 theorems, 127 equations, 6 figures)

This paper contains 50 sections, 40 theorems, 127 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Automorphism groups and distortion
  3. Results
  4. Turing machines and gates
  5. Turing machines
  6. Gates
  7. Definitions
  8. General notions
  9. Subshifts and cellular automata
  10. Turing machines
  11. The SMART machine on cyclic tapes
  12. Action of SMART on cyclic tapes
  13. Encoding SMART configurations
  14. Analysis of SMART configurations
  15. Encoding cyclic configurations into their orbit positions in C_l
  16. ...and 35 more sections

Key Result

Theorem A

For any non-trivial alphabet $A$, the group $\mathrm{Aut}(A^\mathbb{Z})$ has an element $g$ of infinite order such that $|g^n|_F = O(\log^4 n)$ for some finite set $F$.

Figures (6)

  • Figure 1: Bottom-up analysis of SMART configurations: $k \to k+1$
  • Figure 2: Encoding SMART configurations: $F_{\mathrm{init}}$
  • Figure 3: Encoding SMART configurations: $F_{k \to k+1}$ (Part 1: $k \to k+1$)
  • Figure 4: Encoding SMART configurations: $F_{k \to k+1}$ (Part 2: special $\to k+1$)
  • Figure 5: Final encoding step of SMART configurations: $F_{\ell,\mathrm{final}}$
  • ...and 1 more figures

Theorems & Definitions (96)

  • Theorem A
  • Theorem B
  • Lemma 1.0
  • Theorem C
  • Theorem D
  • Theorem E
  • Corollary 1.1
  • Remark 3.1
  • Proposition 3.2
  • proof
  • ...and 86 more