Algebraic Extensions in Free Groups

  • Alexei Miasnikov
  • Enric Ventura
  • Pascal Weil
Conference paper
Part of the Trends in Mathematics book series (TM)


The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g., being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Baumslag, A. Miasnikov and V. Remeslennikov, Malnormality is decidable in free groups, Internat. J. Algebra Comput., 9 (1999), no. 6, 687–692.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    G.M. Bergman, Supports of derivations, free factorizations and ranks of fixed subgroups in free groups, Trans. Amer. Math. Soc., 351 (1999), 1531–1550.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J.-C. Birget, S. Margolis, J. Meakin, P. Weil. PSPACE-completeness of certain algorithmic problems on the subgroups of free groups, in ICALP 94 (S. Abiteboul, E. Shamir éd.), Lecture Notes in Computer Science 820 (Springer, 1994) 274–285.Google Scholar
  4. [4]
    J.-C. Birget, S. Margolis, J. Meakin, P. Weil. PSPACE-completeness of certain algorithmic problems on the subgroups of free groups, Theoretical Computer Science 242 (2000) 247–281.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    W. Dicks, E. Ventura, The group fixed by a family of injective endomorphism of a free group, Contemp. Math., 195 (1996), 1–81.MathSciNetGoogle Scholar
  6. [6]
    S. Gersten, On Whitehead’s algorithm, Bull. Am. Math. Soc., 10 (1984), 281–284.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    I. Kapovich and A. Miasnikov, Stallings Foldings and Subgroups of Free Groups, J. Algebra, 248,2 (2002), 608–668.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Lyndon and P. Schupp, Combinatorial group theory, Springer, (1977, reprinted 2001).Google Scholar
  9. [9]
    W. Magnus, A. Karras and D. Solitar, Combinatorial group theory, Dover Publications, New York, (1976).zbMATHGoogle Scholar
  10. [10]
    G.S. Makanin. Equations in a free group, Izvestiya Akad. Nauk SSSR 46 (1982), 1199–1273 (in Russian). (English translation: Math. USSR Izvestiya 21 (1983), 483–546.)zbMATHMathSciNetGoogle Scholar
  11. [11]
    S. Margolis, M. Sapir and P. Weil, Closed subgroups in pro-V topologies and the extension problems for inverse automata, Internat. J. Algebra Comput. 11,4 (2001), 405–445.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    L. Ribes and P.A. Zalesskii, The pro-p topology of a free group and algorithmic problems in semigroups, Internat. J. Algebra Comput. 4 (1994) 359–374.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Roig, E. Ventura, P. Weil, On the complexity of the Whitehead minimization problem, Internat. J. Algebra Comput., to appear.Google Scholar
  14. [14]
    J.-P. Serre, Arbres, amalgames, SL 2, Astérisque 46, Soc. Math. France, (1977). English translation: Trees, Springer Monographs in Mathematics, Springer, (2003).Google Scholar
  15. [15]
    P. Silva and P. Weil, On an algorithm to decide whether a free group is a free factor of another, Theoretical Informatics and Applications, to appear.Google Scholar
  16. [16]
    J.R. Stallings, Topology of finite graphs, Inventiones Math. 71 (1983), 551–565.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    M. Takahasi, Note on chain conditions in free groups, Osaka Math. Journal 3,2 (1951), 221–225.zbMATHMathSciNetGoogle Scholar
  18. [18]
    E.C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc., 28 (1996), 255–263.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    E.C. Turner, private communication, 2005.Google Scholar
  20. [20]
    E. Ventura, On fixed subgroups of maximal rank, Comm. Algebra, 25 (1997), 3361–3375.zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    E. Ventura, Fixed subgroups in free groups: a survey, Contemp. Math., 296 (2002), 231–255.MathSciNetGoogle Scholar
  22. [22]
    P. Weil. Computing closures of finitely generated subgroups of the free group, in Algorithmic problems in groups and semigroups (J.-C. Birget, S. Margolis, J. Meakin, M. Sapir éds.), Birkhäuser, 2000, 289–307.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Alexei Miasnikov
    • 1
    • 2
  • Enric Ventura
    • 3
    • 4
  • Pascal Weil
    • 5
  1. 1.Dept of Mathematics and StatisticsMcGill UniversityMontréalCanada
  2. 2.Dept of Mathematics and Computer ScienceCity University of New YorkNew York
  3. 3.EPSEMUniversitat Politècnica de CatalunyaManresa (Barcelona)Catalonia
  4. 4.Centre de Recerca MatemàticaBellaterra (Barcelona)Catalonia
  5. 5.LaBRIUniversité de Bordeaux and CNRSTalenceFrance

Personalised recommendations