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Periodic points and residual finiteness of automorphism groups of subshifts

Ville Salo

Abstract

If totally periodic points are dense in a subshift $X$, its automorphism group is residually finite. We show a weak converse: if periodic points are not dense in a subshift $X$, then the automorphism group of $X \times Y$ is not residually finite for full shifts $Y$ (and sufficiently full-shift-like subshifts). On the other hand, we show that the automorphism group of a block gluing $\Z^2$-subshift is always locally embeddable in finite groups (thus sofic). Hochman recently constructed a strongly irreducible $\Z^2$-subshift with no periodic points. Combining our result with this example gives a strongly irreducible $\Z^2$-subshift whose automorphism group is not residually finite, which solves a question of Coornaert and Ceccherini-Silberstein.

Periodic points and residual finiteness of automorphism groups of subshifts

Abstract

If totally periodic points are dense in a subshift , its automorphism group is residually finite. We show a weak converse: if periodic points are not dense in a subshift , then the automorphism group of is not residually finite for full shifts (and sufficiently full-shift-like subshifts). On the other hand, we show that the automorphism group of a block gluing -subshift is always locally embeddable in finite groups (thus sofic). Hochman recently constructed a strongly irreducible -subshift with no periodic points. Combining our result with this example gives a strongly irreducible -subshift whose automorphism group is not residually finite, which solves a question of Coornaert and Ceccherini-Silberstein.
Paper Structure (8 sections, 26 theorems, 21 equations, 1 figure)

This paper contains 8 sections, 26 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and conventions
  3. The process
  4. Permutations under a clopen set
  5. Non-residual finiteness of $\mathrm{Aut}(X \times Y)$
  6. The group $\mathbb{Z}^2$
  7. Permutation groups on Cartesian products
  8. Finitely-generated non-residually-finite subgroups of automorphism groups

Key Result

Theorem 1

If $G$ is finitely generated and $X \subset A^G$ does not have dense totally periodic points, then the group $\mathrm{Aut}(X \times B^G)$ is not residually finite for $|B| \geq 2$.

Figures (1)

  • Figure 1: Application of $f_{U, \pi}$ from Example \ref{['ex:fupi']}. Occurrences of $U$ (i.e. $1$-symbols) are highlighted in the preimage. The corresponding areas where $\pi$ was applied are highlighted in the image. Of course, $11$ is fixed by $\pi$, so the pattern stays fixed.

Theorems & Definitions (55)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Theorem 4
  • Corollary 2
  • Definition 2
  • Lemma 1
  • proof
  • ...and 45 more