, Volume 76, Issue 4, pp 961–988 | Cite as

Conjugacy in Baumslag’s Group, Generic Case Complexity, and Division in Power Circuits

  • Volker Diekert
  • Alexei G. Myasnikov
  • Armin Weiß


The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that \(zx z^{-1} =y\) in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag–Solitar group \(\mathbf{{BS}}_{1,2}\) and the Baumslag group \({\mathbf{{G}}}_{1,2}\) (also known as Baumslag–Gersten group). The conjugacy problem in \({\mathbf{{BS}}}_{1,2}\) is complete for the circuit class \(\mathsf {TC}^0\). To the best of our knowledge \({\mathbf{{BS}}}_{1,2}\) is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group \({\mathbf{{G}}}_{1,2}\) is an HNN-extension of \({\mathbf{{BS}}}_{1,2}\). Hence, decidability of the conjugacy problem in \(\mathbf{{G}}_{1,2}\) outside the so-called “black hole” follows from Borovik et al. (Int J Algebra Comput 17(5/6):963–997, 2007). Decidability everywhere is due to Beese. Moreover, he showed exponential time for the set of elements which cannot be conjugated into \(\mathbf{{BS}}_{1,2}\) (Beese 2012). Here we improve Beese’s result in two directions by showing that the conjugacy problem in \({\mathbf{{G}}}_{1,2}\) can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs, conjugacy in \({\mathbf{{G}}}_{1,2}\) can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in \({\mathbf{{G}}}_{1,2}\) by reducing the divisibility problem in power circuits to the conjugacy problem in \({\mathbf{{G}}}_{1,2}\). The complexity of the divisibility problem in power circuits is an open and interesting problem in integer arithmetic. We conjecture that it cannot be solved in elementary time because we can show that it cannot be solved in elementary time by calculating modulo values in power circuits.


Algorithmic group theory Power circuit Generic case complexity Divisibility problem Conjugacy problem  Baumslag group 


  1. 1.
    Baumslag, G.: A non-cyclic one-relator group all of whose finite quotients are cyclic. J. Aust. Math. Soc. 10(3–4), 497–498 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beese, J.: Das Konjugationsproblem in der Baumslag–Gersten–Gruppe. Diploma thesis, Fakultät Mathematik, Universität Stuttgart (2012). (in German) Google Scholar
  3. 3.
    Borovik, A.V., Myasnikov, A.G., Remeslennikov, V.N.: Generic complexity of the conjugacy problem in HNN-extensions and algorithmic stratification of Miller’s groups. Int. J. Algebra Comput. 17(5/6), 963–997 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ceccherini-Silberstein, T., Grigorchuk, R.I., de la Harpe, P.: Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Trudy Matematicheskogo Instituta Imeni V. A. Steklova. Rossiĭskaya Akad. Nauk 224, 68–111 (1999)zbMATHGoogle Scholar
  5. 5.
    Craven, M.J., Jimbo, H.C.: Evolutionary algorithm solution of the multiple conjugacy search problem in groups, and its applications to cryptography. Groups Complex. Cryptol. 4, 135–165 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Diekert, V., Laun, J., Ushakov, A.: Efficient algorithms for highly compressed data: the word problem in Higman’s group is in P. Int. J. Algebra Comput. 22(8), 1–19 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Diekert, V., Myasnikov, A.G., Weiß, A.: Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem. In: Proceedings of ISSAC 2015. ACM Press (2015). arXiv:1501.05579
  8. 8.
    Gersten, S.M.: Dehn functions and L1-norms of finite presentations. In: Baumslag, G., Miller III, C.F. (eds.) Algorithms and Classification in Combinatorial Group Theory, pp. 195–225. Springer, Berlin (1992)Google Scholar
  9. 9.
    Gersten, S.M.: Isoperimetric and isodiametric functions of finite presentations. In: Geometric Group Theory, vol. 1 (Sussex, 1991), London Mathematical Society Lecture Note Series, vol. 181, pp. 79–96. Cambridge University Press, Cambridge (1993)Google Scholar
  10. 10.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  11. 11.
    Grigoriev, D., Shpilrain, V.: Authentication from matrix conjugation. Groups Complex. Cryptol. 1, 199–205 (2009)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hesse, W.: Division is in uniform \({\rm TC}^{0}\). In: Orejas, F., Spirakis, P.G., van Leeuwen, P.G. (eds.) ICALP, Lecture Notes in Computer Science, vol. 2076, pp. 104–114. Springer (2001)Google Scholar
  13. 13.
    Hesse, W., Allender, E., Barrington, D.A.M.: Uniform constant-depth threshold circuits for division and iterated multiplication. J. Comput. Syst. Sci. 65, 695–716 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kapovich, I., Miasnikov, A.G., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory and random walks. J. Algebra 264, 665–694 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Average-case complexity and decision problems in group theory. Adv. Math. 190, 343–359 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lyndon, R., Schupp, P.: Combinatorial Group Theory. Classics in Mathematics. Springer, Berlin (2001). First edition 1977zbMATHGoogle Scholar
  17. 17.
    Magnus, W.: Das Identitätsproblem für Gruppen mit einer definierenden Relation. Math. Ann. 106, 295–307 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miller III, C.F.: On Group-Theoretic Decision Problems and Their Classification, Annals of Mathematics Studies, vol. 68. Princeton University Press, Princeton (1971)Google Scholar
  19. 19.
    Myasnikov, A., Shpilrain, V., Ushakov, A.: Group-Based Cryptography. Advanced Courses in Mathematics. CRM, Barcelona; Birkhäuser, Basel (2008)Google Scholar
  20. 20.
    Myasnikov, A.G., Ushakov, A., Won, D.W.: The word problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable. J. Algebra 345, 324–342 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Myasnikov, A.G., Ushakov, A., Won, D.W.: Power circuits, exponential algebra, and time complexity. Int. J. Algebra Comput. 22(6), 3–53 (2012)MathSciNetGoogle Scholar
  22. 22.
    Northshield, S.: Cogrowth of regular graphs. Proc. Am. Math. Soc. 116(1), 203–205 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Papadimitriou, Ch.: Computation Complexity. Addison-Wesley, Reading (1994)Google Scholar
  24. 24.
    Shpilrain, V., Zapata, G.: Combinatorial group theory and public key cryptography. Appl. Algebra Eng. Commun. Comput. 17, 291–302 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Vollmer, H.: Introduction to Circuit Complexity. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  26. 26.
    Woess, W.: Random walks on infinite graphs and groups—a survey on selected topics. Lond. Math. Soc. 26, 1–60 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Woess, W.: Random Walks on Infinite Graphs and Groups. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.FMIUniversität StuttgartStuttgartGermany
  2. 2.Department of MathematicsStevens Institute of TechnologyHobokenUSA

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