Realizability of Subgroups by Subshifts of Finite Type

TL;DR

This paper investigates which subgroups can be represented as stabilizers of configurations in subshifts of finite type, linking realizability to properties like aperiodicity and subgroup decidability.

Contribution

It establishes criteria for subgroup realizability in SFTs, introduces periodically rigid groups, and conjectures their classification among finitely generated groups.

Findings

Finitely generated normal subgroup realizability linked to strongly aperiodic SFTs.

Subgroup membership problem must be decidable for realizable subgroups.

Virtually nilpotent and polycyclic groups satisfy the introduced conjecture.

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.