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Mathematische Zeitschrift

, Volume 260, Issue 1, pp 25–30 | Cite as

Subgroup separability in residually free groups

  • Martin R. Bridson
  • Henry Wilton
Article

Abstract

We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type \(\mathrm{FP}_\infty\) are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.

Keywords

Free Group Direct Product Surface Group Nilpotent Group Limit Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Baumslag B. (1967). Residually free groups. Proc. Lond. Math. Soc. 17(3): 402–418 zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baumslag G., Myasnikov A. and Remeslennikov V. (1999). Algebraic geometry over groups. I. Algebraic sets and ideal theory. J. Algebra 219(1): 16–79 zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bridson, M.R., Miller C.F. III.: Structure and finiteness properties of subdirect products of groups. Preprint, http://www.ma.ic.ac.uk/~mbrids/papers/BMiller07/; arXiv:0708.4331
  4. 4.
    Bridson, M.R., Howie, J., Miller C.F. III., Short, H.: Subgroups of direct products of limit groups. Preprint, 2007, arXiv:0704.3935Google Scholar
  5. 5.
    Bridson, M.R., Howie, J., Miller C.F. III., Short, H.: Finitely presented, residually free groups, in preparationGoogle Scholar
  6. 6.
    Kharlampovich O. and Myasnikov A. (1998). Irreducible affine varieties over a free group. I. Irreducibility of quadratic equations and Nullstellensatz. J. Algebra 200: 472–516 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kharlampovich O. and Myasnikov A. (1998). Irreducible affine varieties over a free group. II. Systems in triangular quasi-quadratic form and description of residually free groups. J. Algebra 200: 517–570 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Long, D.D., Reid, A.W.: Subgroup separability and virtual retractions of groups. Topology (to appear)Google Scholar
  9. 9.
    Sela, Z.: Diophantine geometry over groups. I. Makanin-Razborov diagrams. Publ. Math. Inst. Hautes Études Sci. pp 31–105 (2001)Google Scholar
  10. 10.
    Sela Z. (2003). Diophantine geometry over groups. II. Completions, closures and formal solutions. Israel J. Math. 134: 173–254 zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Stallings J.R. (1983). Topology of finite graphs. Invent. Math. 71: 551–565 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Stallings J. (1963). A finitely presented group whose 3-dimensional integral homology is not finitely generated. Am. J. Math. 85: 541–543 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Wilton, H.: Hall’s Theorem for limit groups. GAFA (to appear)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Mathematics Department, Imperial College LondonLondonUK
  2. 2.Department of MathematicsAustinUSA

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