Document Open Access Logo

Subgroup Membership in GL(2,Z)

Author Markus Lohrey

PDF

File

Thumbnail PDF
PDF
LIPIcs.STACS.2021.51.pdf
  • Filesize: 0.8 MB
  • 17 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • free groups
  • virtually free groups
  • subgroup membership
  • matrix groups

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

It is shown that the subgroup membership problem for a virtually free group can be decided in polynomial time where all group elements are represented by so-called power words, i.e., words of the form p_1^{z_1} p_2^{z_2} ⋯ p_k^{z_k}. Here the p_i are explicit words over the generating set of the group and all z_i are binary encoded integers. As a corollary, it follows that the subgroup membership problem for the matrix group GL(2,ℤ) can be decided in polynomial time when all matrix entries are given in binary notation.

Cite As Get BibTex

Markus Lohrey. Subgroup Membership in GL(2,Z). In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.STACS.2021.51

Author Details

Markus Lohrey
  • Universität Siegen, Germany

Funding

  • Lohrey, Markus: Funded by DFG project LO 748/12-1.

References

  1. Jürgen Avenhaus and Klaus Madlener. The Nielsen reduction and P-complete problems in free groups. Theoretical Computer Science, 32(1-2):61-76, 1984. Google Scholar
  2. Jürgen Avenhaus and Dieter Wißmann. Using rewriting techniques to solve the generalized word problem in polycyclic groups. In Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, ISSAC 1989, pages 322-337. ACM Press, 1989. Google Scholar
  3. Frédérique Bassino, Ilya Kapovich, Markus Lohrey, Alexei Miasnikov, Cyril Nicaud, Andrey Nikolaev, Igor Rivin, Vladimir Shpilrain, Alexander Ushakov, and Pascal Weil. Compression techniques in group theory. In Complexity and Randomness in Group Theory, chapter 4. De Gruyter, 2020. Google Scholar
  4. Paul C. Bell, Mika Hirvensalo, and Igor Potapov. The identity problem for matrix semigroups in SL₂(ℤ) is NP-complete. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 187-206. SIAM, 2017. Google Scholar
  5. Michèle Benois. Parties rationnelles du groupe libre. Comptes rendus hebdomadaires des séances de l'Académie des sciences, Séries A, 269:1188-1190, 1969. Google Scholar
  6. Michèle Benois and Jacques Sakarovitch. On the complexity of some extended word problems defined by cancellation rules. Information Processing Letters, 23(6):281-287, 1986. Google Scholar
  7. Michaël Cadilhac, Dmitry Chistikov, and Georg Zetzsche. Rational subsets of Baumslag-Solitar groups. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, volume 168 of LIPIcs, pages 116:1-116:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  8. Volker Diekert and Murray Elder. Solutions of twisted word equations, EDT0L languages, and context-free groups. CoRR, abs/1701.03297, 2017. URL: http://arxiv.org/abs/1701.03297.
  9. Volker Diekert, Igor Potapov, and Pavel Semukhin. Decidability of membership problems for flat rational subsets of GL(2, Q) and singular matrices. In Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, ISSAC 2020, pages 122-129. ACM, 2020. Google Scholar
  10. Stefan Friedl and Henry Wilton. The membership problem for 3-manifold groups is solvable. Algebraic & Geometric Topology, 16(4):1827-1850, 2016. Google Scholar
  11. Zeph Grunschlag. Algorithms in Geometric Group Theory. PhD thesis, University of California at Berkley, 1999. Google Scholar
  12. Yuri Gurevich and Paul E. Schupp. Membership problem for the modular group. SIAM Journal on Computing, 37(2):425-459, 2007. Google Scholar
  13. Artur Jeż. The complexity of compressed membership problems for finite automata. Theory of Computing Systems, 55(4):685-718, 2014. Google Scholar
  14. Ilya Kapovich and Alexei Myasnikov. Stallings foldings and subgroups of free groups. Journal of Algebra, 248(2):608-668, 2002. Google Scholar
  15. Ilya Kapovich, Richard Weidmann, and Alexei Myasnikov. Foldings, graphs of groups and the membership problem. International Journal of Algebra and Computation, 15(1):95-128, 2005. Google Scholar
  16. Olga G. Kharlampovich, Alexei G. Myasnikov, Vladimir N. Remeslennikov, and Denis E. Serbin. Subgroups of fully residually free groups: algorithmic problems. In Group theory, statistics, and cryptography, volume 360 of Contemporary Mathematics, pages 63-101. AMS, Providence, RI, 2004. Google Scholar
  17. Evgeny I. Khukhro and Victor D. Mazurov. Unsolved problems in group theory. the Kourovka notebook. CoRR, arXiv:1401.0300v19, 2020. Problem 12.50. URL: http://arxiv.org/abs/1401.0300v19.
  18. Sang-Ki Ko, Reino Niskanen, and Igor Potapov. On the identity problem for the special linear group and the Heisenberg group. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, volume 107 of LIPIcs, pages 132:1-132:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  19. Klaus-Jörn Lange and Pierre McKenzie. On the complexity of free monoid morphisms. In Proceedings of the 9th International Symposium on Algorithms and Computation, ISAAC 1998, number 1533 in Lecture Notes in Computer Science, pages 247-256. Springer, 1998. Google Scholar
  20. Markus Lohrey. The Compressed Word Problem for Groups. SpringerBriefs in Mathematics. Springer, 2014. Google Scholar
  21. Markus Lohrey and Armin Weiß. The power word problem. In Proceedings of the 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, volume 138 of LIPIcs, pages 43:1-43:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. Google Scholar
  22. M. Lothaire. Combinatorics on Words. Cambridge University Press, 1997. Google Scholar
  23. Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer, 1977. Google Scholar
  24. Anatolij I. Mal'cev. On homomorphisms onto finite groups. American Mathematical Society Translations, Series 2, 119:67-79, 1983. Translation from Ivanov. Gos. Ped. Inst. Ucen. Zap. 18 (1958) 49-60. Google Scholar
  25. Luda Markus-Epstein. Stallings foldings and subgroups of amalgams of finite groups. International Journal of Algebra and Compution, 17(8):1493-1535, 2007. Google Scholar
  26. K. A. Mihaĭlova. The occurrence problem for direct products of groups. Math. USSR Sbornik, 70:241-251, 1966. English translation. Google Scholar
  27. Igor Potapov and Pavel Semukhin. Decidability of the membership problem for 2 times 2 integer matrices. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 170-186. SIAM, 2017. Google Scholar
  28. Eliyahu Rips. Subgroups of small cancellation groups. Bulletin of the London Mathematical Society, 14:45-47, 1982. Google Scholar
  29. Nikolai S. Romanovskiĭ. Some algorithmic problems for solvable groups. Algebra i Logika, 13(1):26-34, 1974. Google Scholar
  30. Nikolai S. Romanovskiĭ. The occurrence problem for extensions of abelian groups by nilpotent groups. Sibirskii Matematicheskii Zhurnal, 21:170-174, 1980. Google Scholar
  31. Paul E. Schupp. Coxeter groups, 2-completion, perimeter reduction and subgroup separability. Geometriae Dedicata, 96:179-198, 2003. Google Scholar
  32. Jean-Pierre Serre. Trees. Springer, 1980. Google Scholar
  33. Charles C. Sims. Computation with permutation groups. In Proceedings of SYMSAC 1971, pages 23-28. Association for Computing Machinery, 1971. Google Scholar
  34. John R. Stallings. Topology of finite graphs. Inventiones Mathematicae, 71(3):551-565, 1983. Google Scholar
  35. Nicholas W. M. Touikan. A fast algorithm for Stallings' folding process. International Journal of Algebra and Computation, 16(6):1031-1045, 2006. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

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