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Algebraic Extensions in Free Groups

  • Alexei Miasnikov
  • Enric Ventura
  • Pascal Weil
Conference paper
  • 727 Downloads
Part of the Trends in Mathematics book series (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.

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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

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