Stability properties for subgroups generated by return words

TL;DR

This paper introduces a stability concept for subgroups generated by return words in shift spaces, unifying and generalizing previous results, and exploring properties like decidability and closure.

Contribution

It proposes a new stability framework for subgroups generated by return words, unifying various prior results and extending their scope.

Findings

Revisits and generalizes existing results on return words

Studies decidability and closure properties of stable subgroups

Provides a unified approach to diverse applications in shift spaces

Abstract

Return words are a classical tool for studying shift spaces with low factor complexity. In recent years, their projection inside groups have attracted some attention, for instance in the context of dendric shift spaces, of generation of pseudorandom numbers (through the welldoc property), and of profinite invariants of shift spaces. Aiming at unifying disparate works, we introduce a notion of stability for subgroups generated by return words. Within this framework, we revisit several existing results and generalize some of them. We also study general aspects of stability, such as decidability or closure under certain operations.