[Paper Review] Boundary action of automaton groups without singular points and Wang tilings

Daniele D’Angeli, Thibault Godin+3 more|arXiv (Cornell University)|Apr 26, 2016

Cellular Automata and Applications17 references3 citations

TL;DR

This paper investigates automaton groups without singular points—points where the stabilizer map on the boundary fails to be continuous—offering a characterization of such groups via contracting automata and bireversible structures. It establishes a deep link between the non-existence of singular points and aperiodic Wang tilings through commuting pairs and helix graphs, proving that elementary-free automata with no non-trivial commuting pairs yield aperiodic tilings.

ABSTRACT

We study automaton groups without singular points, that is, points in the boundary for which the map that associates to each point its stabilizer, is not continuous. This is motivated by the problem of finding examples of infinite bireversible automaton groups with all trivial stabilizers in the boundary, raised by Grigorchuk and Savchuk. We show that, in general, the set of singular points has measure zero. Then we focus our attention on several classes of automata. We characterize those contracting automata generating groups without singular points, and apply this characterization to the Basilica group. We prove that potential examples of reversible automata generating infinite groups without singular points are necessarily bireversible. Then we provide some necessary conditions for such examples to exist, and study some dynamical properties of their Schreier graphs in the boundary. Finally we relate some of those automata with aperiodic tilings of the discrete plane via Wang tilings. This has a series of consequences from the algorithmic and dynamical points of view, and is related to a problem of Gromov regarding the searching for examples of CAT(0) complexes whose fundamental groups are not hyperbolic and contain no subgroup isomorphic to $\mathbb{Z}^{2}$.

Motivation & Objective

• To address Grigorchuk and Savchuk's open problem on the existence of infinite bireversible automaton groups with trivial stabilizers on the boundary.
• To characterize automaton groups without singular points, particularly in the contracting and bireversible classes.
• To establish a connection between the non-existence of singular points and aperiodic Wang tilings via commuting pairs and helix graphs.
• To investigate the dynamical properties of Schreier graphs in the boundary and their relation to group actions.
• To explore decidability and structural constraints for such groups, especially in the context of non-elementary commuting pairs.

Proposed method

• Uses the boundary action of automaton groups and analyzes the continuity of the stabilizer map to define singular points.
• Applies the concept of helix graphs to reduce the problem of singular points to the existence of non-elementary commuting pairs of words.
• Introduces the notion of strongly-singular and essentially-singular helix graphs to characterize the absence of singular points.
• Establishes a correspondence between automaton groups and Wang tilesets via the dual automaton construction, particularly M ⊔ M⁻¹.
• Leverages results from B"uchi automata and language theory to characterize stable automata in the contracting case.
• Uses the Kari-Papasoglu tiling framework to relate periodic vs. aperiodic tilings to the existence of commuting pairs.

Experimental results

Research questions

• RQ1 Under what conditions do automaton groups generated by contracting automata have no singular points?
• RQ2 Can bireversible automaton groups without singular points exist, and what structural constraints must they satisfy?
• RQ3 What is the relationship between non-elementary commuting pairs and the existence of aperiodic Wang tilings?
• RQ4 Is the existence of singular points decidable for a given Mealy automaton?
• RQ5 Can the Schreier graphs of singular points be isomorphic even when the points are distinct?

Key findings

• The set of singular points in the boundary of any automaton group has measure zero.
• For bireversible automaton groups, singular points coincide exactly with points having non-trivial stabilizers, confirming that such groups act essentially freely.
• In the contracting case, automaton groups without singular points are characterized as those generating stable automata, with the language of stable words recognized by B"uchi automata.
• The Basilica group is shown to have no singular points, contrasting with the Hanoi Towers group.
• The existence of a non-elementary commuting pair u ∈ (Q ∖ {e})*, v ∈ Σ* is equivalent to the existence of a periodic tiling of Kari-Papasoglu type.
• A tileset derived from M ⊔ M⁻¹ admits an aperiodic tiling if and only if the corresponding automaton has no non-elementary commuting pairs and is elementary-free, establishing a full characterization via Theorem 6.16.