Aperiodic subshifts of finite type on groups which are not finitely generated

TL;DR

This paper constructs an example of a non-finitely generated group with a strongly aperiodic subshift of finite type and characterizes such groups based on their subgroups and conjugacy class roots.

Contribution

It introduces the first known example of a non-finitely generated group with a strongly aperiodic SFT and provides a complete characterization of these groups.

Findings

Existence of a non-finitely generated group with a strongly aperiodic SFT

Characterization of groups with this property via subgroups and conjugacy classes

Complete classification of such groups

Abstract

We provide an example of a non-finitely generated group which admits a nonempty strongly aperiodic SFT. Furthermore, we completely characterize the groups with this property in terms of their finitely generated subgroups and the roots of their conjugacy classes.