Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shifts on the lamplighter group

Laurent Bartholdi, Ville Salo

TL;DR

This work develops a robust simulation framework for subshifts on the lamplighter group ${\mathcal L}=({\mathbb Z}/2)\wr{\mathbb Z}$ by combining substitutional ${\mathcal S}$-shifts, Wang tilings, and a fixed-point tiling construction. It shows that effective ${\mathcal H}$-systems pull back to sofic ${\mathcal L}$-shifts, and that pullbacks of Cantor systems admit SFT covers, yielding deep consequences: undecidability of the domino problem on ${\mathcal L}$, the existence of strongly aperiodic SFTs, and the precise description of entropy values as upper semicomputable real numbers. The paper further develops a self-similar tiling framework to encode arbitrary effective subshifts, constructs a zero-entropy, strictly ergodic SFT ${X}_{\mathrm{tree}}$ on ${\mathcal L}$, and shows inter-simulability with Baumslag–Solitar groups, thereby transferring the complexity and dynamical properties across related groups. Collectively, these results advance the understanding of symbolic dynamics on nonamenable and locally finite groups and demonstrate how fixed-point and induction techniques yield sharp, computability-theoretic conclusions about SFTs and their entropies.

Abstract

We prove that the lamplighter group admits strongly aperiodic SFTs, has undecidable tiling problem, and the entropies of its SFTs are exactly the upper semicomputable nonnegative real numbers, and some other results. These results follow from two relatively general simulation theorems, which show that for a large class of effective subshifts on the sea-level subgroup, their induction to the lamplighter group is sofic; and the pullback of every effective Cantor system on the integers admits an SFT cover. We exhibit a concrete strongly aperiodic set with $1488$ tetrahedra. We show that metabelian Baumslag-Solitar groups are intersimulable with lamplighter groups, and thus we obtain the same characterization for their entropies.

Shifts on the lamplighter group

TL;DR

This work develops a robust simulation framework for subshifts on the lamplighter group by combining substitutional -shifts, Wang tilings, and a fixed-point tiling construction. It shows that effective -systems pull back to sofic -shifts, and that pullbacks of Cantor systems admit SFT covers, yielding deep consequences: undecidability of the domino problem on , the existence of strongly aperiodic SFTs, and the precise description of entropy values as upper semicomputable real numbers. The paper further develops a self-similar tiling framework to encode arbitrary effective subshifts, constructs a zero-entropy, strictly ergodic SFT on , and shows inter-simulability with Baumslag–Solitar groups, thereby transferring the complexity and dynamical properties across related groups. Collectively, these results advance the understanding of symbolic dynamics on nonamenable and locally finite groups and demonstrate how fixed-point and induction techniques yield sharp, computability-theoretic conclusions about SFTs and their entropies.

Abstract

We prove that the lamplighter group admits strongly aperiodic SFTs, has undecidable tiling problem, and the entropies of its SFTs are exactly the upper semicomputable nonnegative real numbers, and some other results. These results follow from two relatively general simulation theorems, which show that for a large class of effective subshifts on the sea-level subgroup, their induction to the lamplighter group is sofic; and the pullback of every effective Cantor system on the integers admits an SFT cover. We exhibit a concrete strongly aperiodic set with tetrahedra. We show that metabelian Baumslag-Solitar groups are intersimulable with lamplighter groups, and thus we obtain the same characterization for their entropies.
Paper Structure (25 sections, 37 theorems, 41 equations, 3 figures)

This paper contains 25 sections, 37 theorems, 41 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Strategy
  3. Definitions and notations
  4. The lamplighter group
  5. Subshifts, induction and pullback
  6. Subshifts on locally finite groups
  7. Almost odometric subshifts
  8. Subshifts on the lamplighter group
  9. Marking corooted binary trees
  10. A simple ${\mathcal{S}}$-aperiodic tile set
  11. Aperiodic tilesets and undecidability
  12. From effective subshifts to sofic shifts
  13. From Wang tilesets to sofic shifts
  14. From effective subshifts to Wang tilesets
  15. The fixed point construction
  16. ...and 10 more sections

Key Result

Theorem 2

Let $X$ be an effective ${\mathbb Z}$-system. Then $\phi^*(X)$ admits an SFT cover.

Figures (3)

  • Figure 1: A tetrahedron in ${\mathscr L}$, with in blue the "sea level" grid ${\mathcal{S}}={\mathcal{H}}\times{\mathcal{V}}$; it is represented with ${\mathcal{H}}$ going across the page and ${\mathcal{V}}$ going into it.
  • Figure 2: The tetrahedra labellings (first component omitted)
  • Figure 3: A configuration in $T$. The generators $a$ and $b$ of $\mathrm{BS}(1,2)$ are coloured respectively red and blue, and the simulated generators $a,b$ of ${\mathcal{L}}$ are coloured respectively in green and blue.

Theorems & Definitions (80)

  • Theorem 2: = Theorem \ref{['thm:effectiveZ2']}
  • Theorem 3: $\subseteq$ Theorem \ref{['thm:effectiveH2']}
  • Corollary 4: = Corollary \ref{['cor:undecidable2']}
  • Corollary 5: = Corollaries \ref{['cor:examples2']}, \ref{['cor:examples3']}, \ref{['cor:examples4']} and \ref{['cor:examples5']}
  • Theorem 6: = Theorem \ref{['thm:substitutive2']}
  • Definition 3.1: Substitutional shifts
  • Lemma 3.2
  • proof
  • Example 3.3: "Sunny-side-up"
  • Example 3.4: "Thue-Morse"
  • ...and 70 more