A polynomial bound on solutions of quadratic equations in free groups



We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solution sets of quadratic equations in a free group.


  1. 1.
    R. I. Grigorchuk and P. F. Kurchanov, “Some Questions of Group Theory Related to Geometry,” in Algebra-7 (VINITI, Moscow, 1990), Itogi Nauki Tekh., Ser.: Sovrem. Probl. Mat., Fundam. Napravl. 58, pp. 191–256; Engl. transl. in Algebra VII: Combinatorial Group Theory. Applications to Geometry (Springer, Berlin, 1993), Encycl. Math. Sci. 58, pp. 167–232.Google Scholar
  2. 2.
    H. Seifert and W. Threlfall, A Textbook of Topology (Academic Press, New York, 1980).MATHGoogle Scholar
  3. 3.
    R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory (Springer, Berlin, 1977).MATHGoogle Scholar
  4. 4.
    A. I. Mal’cev, “On the Equation zxyx −1 y −1 z −1 = aba −1 b −1 in a Free Group,” Algebra Logika 1(5), 45–50 (1962).MathSciNetGoogle Scholar
  5. 5.
    G. S. Makanin, “Equations in a Free Group,” Izv. Akad. Nauk SSSR, Ser. Mat. 46(6), 1199–1273 (1982) [Math. USSR, Izv. 21, 483–546 (1983)].MathSciNetMATHGoogle Scholar
  6. 6.
    A. Yu. Ol’shanskii, “Homomorphism Diagrams of Surface Groups,” Sib. Mat. Zh. 30(6), 150–171 (1989) [Sib. Math. J. 30, 961–979 (1989)].MathSciNetGoogle Scholar
  7. 7.
    A. A. Razborov, “On Equations in a Free Group,” Candidate (Phys.-Math.) Dissertation (Steklov Math. Inst., Acad. Sci. USSR, Moscow, 1987).Google Scholar
  8. 8.
    D. Bormotov, R. Gilman, and A. Myasnikov, “Solving One-Variable Equations in Free Groups,” J. Group Theory 12(2), 317–330 (2009).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    G. Baumslag, A. Myasnikov, and V. Remeslennikov, “Algebraic Geometry over Groups. I: Algebraic Sets and Ideal Theory,” J. Algebra 219(1), 16–79 (1999).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    I. Bumagin, O. Kharlampovich, and A. Miasnikov, “Isomorphism Problem for Finitely Generated Fully Residually Free Groups,” J. Pure Appl. Algebra 208, 961–977 (2007).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    L. P. Comerford, Jr., “Quadratic Equations over Small Cancellation Groups,” J. Algebra 69(1), 175–185 (1981).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    L. P. Comerford, Jr. and C. C. Edmunds, “Quadratic Equations over Free Groups and Free Products,” J. Algebra 68(2), 276–297 (1981).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    L. P. Comerford, Jr. and C. C. Edmunds, “Solutions of Equations in Free Groups,” in Group Theory: Proc. Conf., Singapore, 1987 (W. de Gruyter, Berlin, 1989), pp. 347–356.Google Scholar
  14. 14.
    M. Culler, “Using Surfaces to Solve Equations in Free Groups,” Topology 20(2), 133–145 (1981).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    F. Dahmani and D. Groves, “The Isomorphism Problem for Toral Relatively Hyperbolic Groups,” Publ. Math., Inst. Hautes Étud. Sci. 107, 211–290 (2008).MathSciNetMATHGoogle Scholar
  16. 16.
    F. Dahmani and V. Guirardel, “The Isomorphism Problem for All Hyperbolic Groups,” Geom. Funct. Anal. 21(2), 223–300 (2011); arXiv: 1002.2590v2.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    V. Diekert and J. M. Robson, “Quadratic Word Equations,” in Jewels Are Forever (Springer, Berlin, 1999), pp. 314–326.CrossRefGoogle Scholar
  18. 18.
    R. I. Grigorchuk and P. F. Kurchanov, “On Quadratic Equations in Free Groups,” in Algebra: Proc. Int. Conf. Dedicated to the Memory of A.I. Mal’cev, Novosibirsk, 1989 (Am. Math. Soc., Providence, RI, 1992), Part 1, Contemp. Math. 131, pp. 159–171.Google Scholar
  19. 19.
    O. Kharlampovich, I. G. Lysënok, A. G. Myasnikov, and N. W. M. Touikan, “The Solvability Problem for Quadratic Equations over Free Groups Is NP-Complete,” Theory Comput. Syst. 47(1), 250–258 (2010).MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    O. Kharlampovich and A. Myasnikov, “Implicit Function Theorem over Free Groups,” J. Algebra 290(1), 1–203 (2005).MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    O. Kharlampovich and A. Myasnikov, “Irreducible Affine Varieties over a Free Group. I: Irreducibility of Quadratic Equations and Nullstellensatz,” J. Algebra 200(2), 472–516 (1998).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    O. Kharlampovich and A. Myasnikov, “Irreducible Affine Varieties over a Free Group. II: Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups,” J. Algebra 200(2), 517–570 (1998).MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    O. Kharlampovich and A. Myasnikov, Equations and Algorithmic Problems in Groups (IMPA, Rio de Janeiro, 2008), Publicações Matemáticas do IMPA.MATHGoogle Scholar
  24. 24.
    O. Kharlampovich and A. G. Myasnikov, “Equations and Fully Residually Free Groups,” in Combinatorial and Geometric Group Theory: Dortmund and Ottawa-Montreal Conf. (Birkhäuser, Basel, 2010), Trends Math., pp. 203–242.CrossRefGoogle Scholar
  25. 25.
    O. Kharlampovich and A. Myasnikov, “Elementary Theory of Free Non-abelian Groups,” J. Algebra 302(2), 451–552 (2006).MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    R. C. Lyndon, “The Equation a 2 b 2 = c 2 in Free Groups,” Mich. Math. J. 6, 89–95 (1959).MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Z. Sela, “The Isomorphism Problem for Hyperbolic Groups. I,” Ann. Math., Ser. 2, 141(2), 217–283 (1995).MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Z. Sela, “Diophantine Geometry over Groups. I: Makanin-Razborov Diagrams,” Publ. Math., Inst. Hautes Étud. Sci. 93, 31–105 (2001).MathSciNetMATHGoogle Scholar
  29. 29.
    Z. Sela, “Diophantine Geometry over Groups. II: Completions, Closures and Formal Solutions,” Isr. J. Math. 134, 173–254 (2003).MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Z. Sela, “Diophantine Geometry over Groups. III: Rigid and Solid Solutions,” Isr. J. Math. 147, 1–73 (2005).MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Z. Sela, “Diophantine Geometry over Groups. IV: An Iterative Procedure for Validation of a Sentence,” Isr. J. Math. 143, 1–130 (2004).MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Z. Sela, “Diophantine Geometry over Groups. V1: Quantifier Elimination. I,” Isr. J. Math. 150, 1–197 (2005).MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Z. Sela, “Diophantine Geometry over Groups. V2: Quantifier Elimination. II,” Geom. Funct. Anal. 16(3), 537–706 (2006).MathSciNetMATHGoogle Scholar
  34. 34.
    Z. Sela, “Diophantine Geometry over Groups. VI: The Elementary Theory of a Free Group,” Geom. Funct. Anal. 16(3), 707–730 (2006).MathSciNetMATHGoogle Scholar
  35. 35.
    M. J. Wicks, “A General Solution of Binary Homogeneous Equations over Free Groups,” Pac. J. Math. 41, 543–561 (1972).MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Department of Mathematical Sciences, Stevens Institute of TechnologyCastle Point on HudsonHobokenUSA

Personalised recommendations