Rational Subsets of Baumslag-Solitar Groups

Authors Michaël Cadilhac , Dmitry Chistikov , Georg Zetzsche

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Michaël Cadilhac
  • DePaul University, Chicago, IL, USA
Dmitry Chistikov
  • Centre for Discrete Mathematics and its Applications (DIMAP) & Department of Computer Science, University of Warwick, Coventry, UK
Georg Zetzsche
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany


The research for this work was carried out in part at the Autobóz Research Camp in 2019 in Firbush, Scotland. The authors are very grateful to Piotr Hofman for comments that led to important insights and to Michael Blondin as well as to all other participants for discussions.

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Michaël Cadilhac, Dmitry Chistikov, and Georg Zetzsche. Rational Subsets of Baumslag-Solitar Groups. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 116:1-116:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 GL(2,ℚ). We show that rational subset membership for Baumslag-Solitar groups BS(1,q) with q ≥ 2 is decidable and PSPACE-complete. To this end, we introduce a word representation of the elements of BS(1,q): their pointed expansion (PE), an annotated q-ary expansion. Seeing subsets of BS(1,q) as word languages, this leads to a natural notion of PE-regular subsets of BS(1,q): these are the subsets of BS(1,q) whose sets of PE are regular languages. Our proof shows that every rational subset of BS(1,q) is PE-regular. Since the class of PE-regular subsets of 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 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 BS(1,q) has finite index.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Models of computation
  • Rational subsets
  • Baumslag-Solitar groups
  • decidability
  • regular languages
  • pointed expansion


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