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Realizability of Subgroups by Subshifts of Finite Type

Nicolás Bitar

TL;DR

This work develops a comprehensive framework for realizing subgroups as stabilizers of configurations in subshifts of finite type, linking subgroup realizability to the existence of strongly aperiodic SFTs on quotients and establishing algorithmic consequences. It introduces canonical constructions (free extension, higher power, pull-back, push-forward) to transfer SFT properties across groups and their quotients, enabling precise control of stabilizers. A central result is that a nontrivial finitely generated normal subgroup $N\trianglelefteq G$ is realizable if and only if $G/N$ admits a strongly aperiodic SFT, with broader implications for decidability of subgroup membership. The paper then develops the notion of periodic rigidity, conjecturing a sharp classification: finitely generated periodically rigid groups are exactly the virtually $\mathbb{Z}$ groups or torsion-free virtually $\mathbb{Z}^2$ groups, and proves this conjecture for virtually nilpotent and polycyclic groups, advancing understanding of how large-scale group structure governs aperiodicity phenomena in symbolic dynamics.

Abstract

We study the problem of realizing families of subgroups as the set of stabilizers of configurations from a subshift of finite type (SFT). This problem generalizes both the existence of strongly and weakly aperiodic SFTs. We show that a finitely generated normal subgroup is realizable if and only if the quotient by the subgroup admits a strongly aperiodic SFT. We also show that if a subgroup is realizable, its subgroup membership problem must be decidable. The article also contains the introduction of periodically rigid groups, which are groups for which every weakly aperiodic subshift of finite type is strongly aperiodic. We conjecture that the only finitely generated periodically rigid groups are virtually $\mathbb{Z}$ groups and torsion-free virtually $\mathbb{Z}^2$ groups. Finally, we show virtually nilpotent and polycyclic groups satisfy the conjecture.

Realizability of Subgroups by Subshifts of Finite Type

TL;DR

This work develops a comprehensive framework for realizing subgroups as stabilizers of configurations in subshifts of finite type, linking subgroup realizability to the existence of strongly aperiodic SFTs on quotients and establishing algorithmic consequences. It introduces canonical constructions (free extension, higher power, pull-back, push-forward) to transfer SFT properties across groups and their quotients, enabling precise control of stabilizers. A central result is that a nontrivial finitely generated normal subgroup is realizable if and only if admits a strongly aperiodic SFT, with broader implications for decidability of subgroup membership. The paper then develops the notion of periodic rigidity, conjecturing a sharp classification: finitely generated periodically rigid groups are exactly the virtually groups or torsion-free virtually groups, and proves this conjecture for virtually nilpotent and polycyclic groups, advancing understanding of how large-scale group structure governs aperiodicity phenomena in symbolic dynamics.

Abstract

We study the problem of realizing families of subgroups as the set of stabilizers of configurations from a subshift of finite type (SFT). This problem generalizes both the existence of strongly and weakly aperiodic SFTs. We show that a finitely generated normal subgroup is realizable if and only if the quotient by the subgroup admits a strongly aperiodic SFT. We also show that if a subgroup is realizable, its subgroup membership problem must be decidable. The article also contains the introduction of periodically rigid groups, which are groups for which every weakly aperiodic subshift of finite type is strongly aperiodic. We conjecture that the only finitely generated periodically rigid groups are virtually groups and torsion-free virtually groups. Finally, we show virtually nilpotent and polycyclic groups satisfy the conjecture.
Paper Structure (25 sections, 50 theorems, 44 equations, 2 figures)

This paper contains 25 sections, 50 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the Article
  3. Conventions and Notation
  4. Background and Definitions
  5. Symbolic Dynamics
  6. Group presentations and the word problem
  7. Strongly aperiodic SFTs
  8. Weakly aperiodic SFTs
  9. Canonical constructions
  10. Free extension
  11. Higher power and higher block
  12. Pull-back
  13. Push-forward
  14. Subgroup Realizability: How I Learned to Stop Worrying and Love Periodicity
  15. General properties
  16. ...and 10 more sections

Key Result

Theorem A

Let $N\trianglelefteq G$ be a non-trivial finitely generated normal subgroup. Then, $N$ is realizable in $G$ if and only if $G/N$ admits a strongly aperiodic SFT.

Figures (2)

  • Figure 1: The graph $\Gamma_{n+4}$ that defines the SFT with set of multiples $\{n+4\mid n\in\mathbb{N}\}$.
  • Figure 2: For SFTs in $\mathbb{Z}^2$, having a configuration with non-trivial stabilizer (on the left) implies the existence of a periodic configuration (on the right). This is done by finding a repeating motif on the strip defined by the period vector, and repeating this motif in a way compatible with the forbidden patterns of the nearest neighbor SFT.

Theorems & Definitions (109)

  • Theorem A
  • Theorem B
  • Conjecture C
  • Theorem D
  • Theorem E
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 99 more