Document Open Access Logo

Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations

Author Ruiwen Dong

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.ICALP.2025.154.pdf
  • Filesize: 0.83 MB
  • 20 pages
HTML5 Logo HTML (experimental)
LIPIcs.ICALP.2025.154.html

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Computing methodologies → Symbolic and algebraic manipulation
Keywords
  • Submonoid Membership
  • lamplighter groups
  • S-unit equations
  • p-automatic sets
  • Knapsack in groups

Metrics

Abstract

We show that Submonoid Membership is decidable in n-dimensional lamplighter groups (ℤ/pℤ) ≀ ℤⁿ for any prime p and integer n. More generally, we show decidability of Submonoid Membership in semidirect products of the form 𝒴 ⋊ ℤⁿ, where 𝒴 is any finitely presented module over the Laurent polynomial ring 𝔽_p[X₁^{±}, …, X_n^{±}]. Combined with a result of Shafrir (2024), this gives the first example of a group G and a finite index subgroup G̃ ≤ G, such that Submonoid Membership is decidable in G̃ but undecidable in G.
To obtain our decidability result, we reduce Submonoid Membership in 𝒴 ⋊ ℤⁿ to solving S-unit equations over 𝔽_p[X₁^{±}, …, X_n^{±}]-modules. We show that the solution set of such equations is effectively p-automatic, extending a result of Adamczewski and Bell (2012). As an intermediate result, we also obtain that the solution set of the Knapsack Problem in 𝒴 ⋊ ℤⁿ is effectively p-automatic.

Cite As Get BibTex

Ruiwen Dong. Submonoid Membership in n-Dimensional Lamplighter Groups and S-Unit Equations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 154:1-154:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.154

Author Details

Ruiwen Dong
  • Department of Mathematics, Saarland University, Saarbrücken, Germany
  • Magdalen College, University of Oxford, UK

Funding

This work was funded by the European Union (ERC, AdG grant 101097307). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

Acknowledgements

The author would like to thank Doron Shafrir for pointing out the open problem on Submonoid Membership in finite extension of groups, as well as for discussion of his work [Shafrir, 2024]. The author would like to thank the anonymous reviewers for their helpful comments.

References

  1. Boris Adamczewski and Jason P. Bell. On vanishing coefficients of algebraic power series over fields of positive characteristic. Inventiones mathematicae, 187(2):343-393, 2012. Google Scholar
  2. László Babai, Robert Beals, Jin-yi Cai, Gábor Ivanyos, and Eugene M. Luks. Multiplicative equations over commuting matrices. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 498-507, 1996. URL: http://dl.acm.org/citation.cfm?id=313852.314109.
  3. Sebastián Barbieri, Jarkko Kari, and Ville Salo. The group of reversible Turing machines. In Cellular Automata and Discrete Complex Systems: 22nd IFIP WG 1.5 International Workshop, AUTOMATA 2016, Zurich, Switzerland, June 15-17, 2016, Proceedings 22, pages 49-62. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-39300-1_5.
  4. Laurent Bartholdi and Ville Salo. Simulations and the lamplighter group. Groups, Geometry, and Dynamics, 16(4):1461-1514, 2022. Google Scholar
  5. Gilbert Baumslag, Frank B. Cannonito, and Charles F. Miller III. Computable algebra and group embeddings. Journal of Algebra, 69(1):186-212, 1981. Google Scholar
  6. Gilbert Baumslag, Frank B. Cannonito, and Derek J.S. Robinson. The algorithmic theory of finitely generated metabelian groups. Transactions of the American Mathematical Society, 344(2):629-648, 1994. Google Scholar
  7. Robert Beals and László Babai. Las Vegas algorithms for matrix groups. In Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science, pages 427-436. IEEE, 1993. URL: https://doi.org/10.1109/SFCS.1993.366844.
  8. Michèle Benois. Parties rationnelles du groupe libre. C. R. Acad. Sci. Paris, Sér. A,, 269:1188-1190, 1969. Google Scholar
  9. Vincent D. Blondel, Emmanuel Jeandel, Pascal Koiran, and Natacha Portier. Decidable and undecidable problems about quantum automata. SIAM Journal on Computing, 34(6):1464-1473, 2005. URL: https://doi.org/10.1137/S0097539703425861.
  10. Corentin Bodart. Membership problems in nilpotent groups. arXiv preprint arXiv:2401.15504, 2024. URL: https://doi.org/10.48550/arXiv.2401.15504.
  11. Michaël Cadilhac, Dmitry Chistikov, and Georg Zetzsche. Rational subsets of Baumslag-Solitar groups. In 47th International Colloquium on Automata, Languages, and Programming, ICALP, volume 168 of LIPIcs, pages 116:1-116:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ICALP.2020.116.
  12. Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On reachability problems for low-dimensional matrix semigroups. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 44:1-44:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.ICALP.2019.44.
  13. Harm Derksen. A Skolem-Mahler-Lech theorem in positive characteristic and finite automata. Inventiones mathematicae, 168(1):175-224, 2007. Google Scholar
  14. Harm Derksen, Emmanuel Jeandel, and Pascal Koiran. Quantum automata and algebraic groups. Journal of Symbolic Computation, 39(3-4):357-371, 2005. URL: https://doi.org/10.1016/j.jsc.2004.11.008.
  15. Ruiwen Dong. Recent advances in algorithmic problems for semigroups. ACM SIGLOG News, 10(4):3-23, 2023. URL: https://doi.org/10.1145/3636362.3636365.
  16. Ruiwen Dong. Linear equations with monomial constraints and decision problems in abelian-by-cyclic groups. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1892-1908. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.59.
  17. David Eisenbud. Commutative algebra: with a view toward algebraic geometry, volume 150. Springer Science & Business Media, 2013. Google Scholar
  18. Moses Ganardi, Daniel König, Markus Lohrey, and Georg Zetzsche. Knapsack problems for wreath products. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, volume 96 of LIPIcs, pages 32:1-32:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPICS.STACS.2018.32.
  19. Rostislav Grigorchuk and Andrzej Żuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geometriae Dedicata, 87(1):209-244, 2001. Google Scholar
  20. Zeph Grunschlag. Algorithms in geometric group theory. University of California, Berkeley, 1999. Google Scholar
  21. Philipp Hieronymi and Christian Schulz. A strong version of Cobham’s theorem. SIAM Journal on Computing, 51(6):1400-1421, 2022. URL: https://doi.org/10.1137/22M1538065.
  22. Ehud Hrushovski, Joël Ouaknine, Amaury Pouly, and James Worrell. Polynomial invariants for affine programs. In Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, pages 530-539, 2018. URL: https://doi.org/10.1145/3209108.3209142.
  23. Mark Kambites, Pedro V. Silva, and Benjamin Steinberg. The spectra of lamplighter groups and Cayley machines. Geometriae Dedicata, 120(1):193-227, 2006. Google Scholar
  24. Toghrul Karimov, Florian Luca, Joris Nieuwveld, Joël Ouaknine, and James Worrell. On the decidability of Presburger arithmetic expanded with powers. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2755-2778. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.89.
  25. Markus Lohrey. Membership problems in infinite groups. In Conference on Computability in Europe, pages 44-59. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-64309-5_5.
  26. Markus Lohrey and Benjamin Steinberg. The submonoid and rational subset membership problems for graph groups. Journal of Algebra, 320(2):728-755, 2008. Google Scholar
  27. Markus Lohrey and Benjamin Steinberg. Tilings and submonoids of metabelian groups. Theory of Computing Systems, 48(2):411-427, 2011. URL: https://doi.org/10.1007/S00224-010-9264-9.
  28. Markus Lohrey, Benjamin Steinberg, and Georg Zetzsche. Rational subsets and submonoids of wreath products. Information and Computation, 243:191-204, 2015. URL: https://doi.org/10.1016/J.IC.2014.12.014.
  29. Russell Lyons, Robin Pemantle, and Yuval Peres. Random walks on the lamplighter group. The annals of probability, pages 1993-2006, 1996. Google Scholar
  30. A. Markov. On an unsolvable problem concerning matrices. Doklady Akad. Nauk SSSR, 78(6):1089-1092, 1951. Google Scholar
  31. K. A. Mikhailova. The entrance problem for direct unions of groups. Doklady Akad. Nauk SSSR, 119(6):1103-1105, 1958. Google Scholar
  32. David Mumford. The Red Book of Varieties and Schemes, volume 1358 of Lecture Notes in Mathematics. Springer, 1999. Google Scholar
  33. Igor Potapov and Pavel Semukhin. Decidability of the membership problem for 2 × 2 integer matrices. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 170-186. SIAM, 2017. Google Scholar
  34. Felix Potthast. Submonoid Membership for Wreath Products. Bachelor’s thesis, Universität Siegen, 2020. Based on the unpublished draft of Doron Shafrir: The Submonoid Problem in Torsion-by-ℤ^d Groups, private communication with Markus Lohrey and Benjamin Steinberg. May 2018. Google Scholar
  35. N.S. Romanovskii. Some algorithmic problems for solvable groups. Algebra and Logic, 13(1):13-16, 1974. Google Scholar
  36. Elizabeth W. Rutman. Gröbner bases and primary decomposition of modules. Journal of symbolic computation, 14(5):483-503, 1992. URL: https://doi.org/10.1016/0747-7171(92)90019-Z.
  37. Doron Shafrir. Is decidability of the Submonoid Membership Problem closed under finite extensions? arXiv preprint arXiv:2405.12921, 2024. URL: https://doi.org/10.48550/arXiv.2405.12921.
  38. Pedro V. Silva and Benjamin Steinberg. On a class of automata groups generalizing lamplighter groups. International journal of algebra and computation, 15(05n06):1213-1234, 2005. URL: https://doi.org/10.1142/S0218196705002761.
  39. Pierre Wolper and Bernard Boigelot. On the construction of automata from linear arithmetic constraints. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pages 1-19. Springer, 2000. URL: https://doi.org/10.1007/3-540-46419-0_1.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact