Skip to main content
SpringerLink
Log in
Book cover

Geometric Group Theory pp 225–253Cite as

Algebraic Extensions in Free Groups

  • Conference paper

Part of the book series: Trends in Mathematics ((TM))

Abstract

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.

This article originates from the Barcelona conference.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Baumslag, A. Miasnikov and V. Remeslennikov, Malnormality is decidable in free groups, Internat. J. Algebra Comput., 9 (1999), no. 6, 687–692.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. W. Dicks, E. Ventura, The group fixed by a family of injective endomorphism of a free group, Contemp. Math., 195 (1996), 1–81.

    MathSciNet  Google Scholar 

  6. S. Gersten, On Whitehead’s algorithm, Bull. Am. Math. Soc., 10 (1984), 281–284.

    Article  MATH  MathSciNet  Google Scholar 

  7. I. Kapovich and A. Miasnikov, Stallings Foldings and Subgroups of Free Groups, J. Algebra, 248,2 (2002), 608–668.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Lyndon and P. Schupp, Combinatorial group theory, Springer, (1977, reprinted 2001).

    Google Scholar 

  9. W. Magnus, A. Karras and D. Solitar, Combinatorial group theory, Dover Publications, New York, (1976).

    MATH  Google Scholar 

  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.)

    MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Roig, E. Ventura, P. Weil, On the complexity of the Whitehead minimization problem, Internat. J. Algebra Comput., to appear.

    Google Scholar 

  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. 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. J.R. Stallings, Topology of finite graphs, Inventiones Math. 71 (1983), 551–565.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Takahasi, Note on chain conditions in free groups, Osaka Math. Journal 3,2 (1951), 221–225.

    MATH  MathSciNet  Google Scholar 

  18. E.C. Turner, Test words for automorphisms of free groups, Bull. London Math. Soc., 28 (1996), 255–263.

    Article  MATH  MathSciNet  Google Scholar 

  19. E.C. Turner, private communication, 2005.

    Google Scholar 

  20. E. Ventura, On fixed subgroups of maximal rank, Comm. Algebra, 25 (1997), 3361–3375.

    Article  MATH  MathSciNet  Google Scholar 

  21. E. Ventura, Fixed subgroups in free groups: a survey, Contemp. Math., 296 (2002), 231–255.

    MathSciNet  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Dept of Mathematics and Statistics, McGill University, Burnside Hall, room 915 805 Sherbrooke West, Montréal, Quebec, Canada, H3A 2K6

    Alexei Miasnikov

  2. Dept of Mathematics and Computer Science, City University of New York, New York

    Alexei Miasnikov

  3. EPSEM, Universitat Politècnica de Catalunya, Av. Bases de Manresa 61-73, 08242, Manresa (Barcelona), Catalonia

    Enric Ventura

  4. Centre de Recerca Matemàtica, Bellaterra (Barcelona), Catalonia

    Enric Ventura

  5. LaBRI, Université de Bordeaux and CNRS, 351 cours de la Libération, F-33400, Talence, France

    Pascal Weil

Authors
  1. Alexei Miasnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Enric Ventura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pascal Weil
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Section de mathématiques, Université de Genève, 2-4, rue du Lièvre, Case postale 64, 1211, Genève 4, Switzerland

    Goulnara N. Arzhantseva

  2. Departament de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya Escola Politècnica Superior de Castelldefels Avda, Del Canal Olímpic s/n, 08860, Castelldefels (Barcelona), Spain

    José Burillo

  3. Institut de Mathématiques B (IMB), École Polytechnique Fédérale de Lausanne (EPFL), 1015, Lausanne, Switzerland

    Laurent Bartholdi

  4. Department of Applied Mathematics III Escola Politècnica Superior d’Enginyeria de Manresa, Universitat Politècnica de Catalunya, Av.Bases de Manresa 61-73, 08242, Manresa / Barcelona, Spain

    Enric Ventura

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Miasnikov, A., Ventura, E., Weil, P. (2007). Algebraic Extensions in Free Groups. In: Arzhantseva, G.N., Burillo, J., Bartholdi, L., Ventura, E. (eds) Geometric Group Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8412-8_12

Download citation

Publish with us

Policies and ethics

Search

Navigation