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SFT covers for actions of the first Grigorchuk group

Rostislav Grigorchuk, Ville Salo

Abstract

We study symbolic dynamical representations of actions of the first Grigorchuk group $G$, namely its action on the boundary of the infinite rooted binary tree, its representation in the topological full group of a minimal substitutive $\mathbb{Z}$-shift, and its representation as a minimal system of Schreier graphs. We show that the first system admits an SFT cover, and the latter two systems are conjugate to sofic subshifts on $G$, but are not of finite type.

SFT covers for actions of the first Grigorchuk group

Abstract

We study symbolic dynamical representations of actions of the first Grigorchuk group , namely its action on the boundary of the infinite rooted binary tree, its representation in the topological full group of a minimal substitutive -shift, and its representation as a minimal system of Schreier graphs. We show that the first system admits an SFT cover, and the latter two systems are conjugate to sofic subshifts on , but are not of finite type.
Paper Structure (20 sections, 38 theorems, 39 equations, 1 table)

This paper contains 20 sections, 38 theorems, 39 equations, 1 table.

Table of Contents

  1. Introduction
  2. Symbolic dynamics
  3. Effectively closed and totally non-free actions
  4. Simulation
  5. Results
  6. Structure of the paper
  7. Acknowledgements
  8. Preliminaries
  9. The group $\mathcal{G}$ and its actions
  10. Action on (the boundary of) a tree
  11. Action on a $\mathbb{Z}$-subshift
  12. Action on Schreier graphs
  13. Basic constructions on SFTs
  14. The union lemma
  15. The finite-index lemma
  16. ...and 5 more sections

Key Result

Theorem 1

Let $G, H, K$ be three finitely-generated infinite groups, and $\pi : G \times H \times K \to G$ the natural projection. Then the $\pi$-pullback of any expansive effective $G$-system admits an $G \times H \times K$-SFT cover.

Theorems & Definitions (77)

  • Remark 1
  • Theorem 1: Ba19
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Proposition 1
  • Lemma 1
  • Definition 1
  • Definition 2
  • Remark 2
  • ...and 67 more