Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T18:42:22.852Z Has data issue: false hasContentIssue false
  • English
  • Français

Decidability problem for exponential equations in finitely presented groups

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  22 November 2022

Oleg Bogopolski  and
Aleksander Ivanov Open the ORCID record for Aleksander Ivanov [Opens in a new window]
Oleg Bogopolski*
Affiliation:
Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland
Aleksander Ivanov
Affiliation:
Department of Applied Mathematics, Silesian University of Technology, ul. Kaszubska 23, 44-101 Gliwice, Poland e-mail: Aleksander.Iwanow@polsl.pl
*
e-mail: Oleg_Bogopolski@yahoo.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the following decision problem: given an exponential equation $a_1g_1^{x_1}a_2g_2^{x_2}\dots a_ng_n^{x_n}=1$ over a recursively presented group G, decide if it has a solution with all $x_i$ in $\mathbb {Z}$. We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.

Keywords

Decidability problemsexponential equationsknapsack problemfinitely presented groups

MSC classification

Primary: 20F10: Word problems, other decision problems, connections with logic and automata 20F70: Algebraic geometry over groups; equations over groups
Secondary: 20F05: Generators, relations, and presentations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 3 , September 2023 , pp. 731 - 748
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bier, A., and Bogopolski, O., Exponential equations in acylindrically hyperbolic groups. Preprint, 2021. arXiv:2106.11385v1 CrossRefGoogle Scholar
Bilanovic, I., Chubb, J., and Roven, S., Detecting properties from presentations of groups . Arch. Math. Log. 59(2020), 293–312.CrossRefGoogle Scholar
Donald, J., Collins, on embedding groups and the conjugacy problem . J. Lond. Math. Soc. (2) 1(1996), 674–682.Google Scholar
Dudkin, F. and Treyer, A., Knapsack problem for Baumslag–Solitar groups . Siberian J. Pure Appl. Math. 18(2018), no. 4, 43–55.Google Scholar
Figelius, M., Ganardi, M., Lohrey, M., and Zetzsche, G., The complexity of knapsack problems in wreath products . In: Proceedings of the 47th international colloquium on automata, languages, and programming, ICALP 2020, Leibniz International Proceedings in Informatics, 168, Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Germany, 2020, pp. 126:1–126:18.Google Scholar
Frenkel, E., Nikolaev, A., and Ushakov, A., Knapsack problems in products of groups . J. Symb. Comput. 76(2016), 96–108.CrossRefGoogle Scholar
Ganardi, M., König, D., Lohrey, M., and Zetzsche, G., Knapsack problems for wreath products . In: Proceedings of STACS (2018), Leibniz International Proceedings in Informatics, 96, Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Germany, 2018, pp. 1–13.Google Scholar
Kharlampovich, O., The word problem for the Burnside varieties . J. Algebra 173(1995), 613–621.CrossRefGoogle Scholar
König, D., Lohrey, M., and Zetzsche, G., Knapsack and subset sum problems for nilpotent, polycyclic, and co-context-free groups . In: Algebra and computer science, Contemporary Mathematics, 677, American Mathematical Society, Providence, RI, 2016, pp. 138–153.Google Scholar
Lohrey, M., Rational subsets of unitriangular groups . Int. J. Algebra Comput. 25(2015), nos. 1–2, 113–121.CrossRefGoogle Scholar
Lohrey, M., Knapsack in hyperbolic groups . J. Algebra 545(2020), no. 1, 390–415.CrossRefGoogle Scholar
Lohrey, M. and Zetzsche, G., Knapsack in graph groups, HNN-extensions and amalgamated products . Theory Comput. Syst. 62(2018), no. 1, 192–246.CrossRefGoogle Scholar
Lohrey, M. and Zetzsche, G., Knapsack and the power word problem in solvable Baumslag–Solitar groups . In: Proceedings of the 45th international symposium on mathematical foundations of computer science, MFCS 2020, Leibniz International Proceedings in Informatics, 170, Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Germany, 2020, pp. 67:1–67:15.Google Scholar
Lyndon, R. C. and Schupp, P. E., Combinatorial group theory, Springer, Berlin, 1977.Google Scholar
McCool, J., Unsolvable problems in groups with solvable word problem . Can. J. Math. 22(1970), 836–838.CrossRefGoogle Scholar
Mishchenko, A. and Treier, A., Knapsack problem for nilpotent groups . Groups Complex. Cryptol. 9(2017), no. 1, 87–98.CrossRefGoogle Scholar
Myasnikov, A., Nikolaev, A., and Ushakov, A., Knapsack problems in groups . Math. Comp. 84(2015), no. 292, 987–1016.CrossRefGoogle Scholar
Ol’shanskii, A. Yu., SQ-universality of hyperbolic groups . Mat. Sb. 186(1995), no. 8, 119–132.Google Scholar
Ol’shanskii, A. Y. and Sapir, M. V., The conjugacy problem and Higman embeddings . Mem. Amer. Math. Soc. 170(2004), no. 804, vii + 131.Google Scholar
Ol’shanskii, A. Y. and Sapir, M. V., Subgroups of finitely presented groups with solvable conjugacy problem . Int. J. Algebra Comput. 15(2005), nos. 5–6, 1–10.Google Scholar
Ol’shanskii, A. Yu. and Sapir, M. V., Algorithmic problems in groups with quadratic Dehn function. Preprint, 2020. arXiv:2012.10417v1 Google Scholar
Soare, R. I., Turing computability. Theory and applications, Springer, Berlin, 2016.Google Scholar