Rational subsets of Baumslag-Solitar groups

TL;DR

This paper proves that the rational subset membership problem for Baumslag-Solitar groups BS(1,q) with q≥2 is decidable and PSPACE-complete, introducing a new word representation called pointed expansion to analyze these subsets.

Contribution

It introduces the PE representation for elements of BS(1,q) and shows all rational subsets are PE-regular, enabling various decision problems to be solved.

Findings

Rational subset membership is decidable and PSPACE-complete.

Every rational subset of BS(1,q) is PE-regular.

Decidability results for emptiness, membership, and recognizability of rational subsets.

Abstract

We consider the rational subset membership problem for Baumslag-Solitar groups. These groups form a prominent class in the area of algorithmic group theory, and they were recently identified as an obstacle for understanding the rational subsets of $\text{GL}(2,\mathbb{Q})$. We show that rational subset membership for Baumslag-Solitar groups $\text{BS}(1,q)$ with $q\ge 2$ is decidable and PSPACE-complete. To this end, we introduce a word representation of the elements of $\text{BS}(1,q)$: their pointed expansion (PE), an annotated $q$-ary expansion. Seeing subsets of $\text{BS}(1,q)$ as word languages, this leads to a natural notion of PE-regular subsets of $\text{BS}(1, q)$: these are the subsets of $\text{BS}(1,q)$ whose sets of PE are regular languages. Our proof shows that every rational subset of $\text{BS}(1,q)$ is PE-regular. Since the class of PE-regular subsets of $\text{BS}(1,q)$ is well-equipped with closure properties, we obtain further applications of these results. Our results imply that (i) emptiness of Boolean combinations of rational subsets is decidable, (ii) membership to each fixed rational subset of $\text{BS}(1,q)$ is decidable in logarithmic space, and (iii) it is decidable whether a given rational subset is recognizable. In particular, it is decidable whether a given finitely generated subgroup of $\text{BS}(1,q)$ has finite index.