Israel Journal of Mathematics

, Volume 95, Issue 1, pp 231–251

Augmented group systems and shifts of finite type

Article

Abstract

Let (G, χ, x) be a triple consisting of a finitely presented groupG, epimorphism χ:GZ, and distinguished elementxG such that χ(x)=1. Given a finite symmetric groupSr, we construct a finite directed graph Γ that describes the set Φr of representations π: Ker χ →Sr as well as the mapping σxr→Φr defined by (σxϱ)(a) = ϱ(x−1ax) for alla ∈ Ker χ. The pair (Φrx has the structure of a shift of finite type, a well-known type of compact 0-dimensional dynamical system. We discuss basic properties and applications of therepresentation shiftrx), including applications to knot theory.

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References

  1. [BaSo] G. Baumslag and D. Solitar,Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society68 (1962), 199–201.MATHMathSciNetGoogle Scholar
  2. [BiNeSt] R. Bieri, W. D. Neumann and R. Strebel,A geometric invariant of discrete groups, Inventiones mathematicae90 (1987), 451–477.MATHCrossRefMathSciNetGoogle Scholar
  3. [BiSt] R. Bieri and R. Strebel,Almost finitely presented soluble groups, Commentarii Mathematici Helvetici53 (1978), 258–278.MATHCrossRefMathSciNetGoogle Scholar
  4. [GoWh] F. González-Acuna and W. Whitten,Imbeddings of 3-manifold groups, Memoirs of the American Mathematical Society474 (1992).Google Scholar
  5. [HaKe] J. C. Hausmann and M. Kervaire,Sous-groupes dérivés des groupes de noeuds, L'Enseignement Mathématique24 (1978), 111–123.MATHMathSciNetGoogle Scholar
  6. [LiMa] D. Lind and B. Marcus,An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.MATHGoogle Scholar
  7. [LySc] R. C. Lyndon and P. E. Schup,Combinatorial Group Theory, Springer-Verlag, Berlin, 1977.MATHGoogle Scholar
  8. [Ra] E. S. Rapaport,Knot-like groups, in Annals of Mathematical Studies84, Princeton University Press, Princeton, 1975, pp. 119–133.Google Scholar
  9. [Re] B. Renz,Geometric invariants and HNN-extensions, inGroup Theory (Proceedings of a Conference in Singapore, 1987), de Gruyter, Berlin, 1989, pp. 465–484.Google Scholar
  10. [Ro] D. Rolfsen,Knots and Links, Mathematics Lecture Series7, Publish or Perish, Inc., Berkeley, 1976.MATHGoogle Scholar
  11. [Si1] D. S. Silver,Augmented group systems and n-knots, Mathematische Annalen296 (1993), 585–593.MATHCrossRefMathSciNetGoogle Scholar
  12. [Si2] D. S. Silver,Knot invariants from topological entropy, Topology and its Applications61 (1995), 159–177.MATHCrossRefMathSciNetGoogle Scholar
  13. [Si3] D. S. Silver,Growth rates of n-knots, Topology and its Applications42 (1991), 217–230.MATHCrossRefMathSciNetGoogle Scholar
  14. [Th] W. P. Thurston,Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bulletin of the American Mathematical Society6 (1982), 357–381.MATHMathSciNetGoogle Scholar

Copyright information

© The Magnes Press 1996

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of South AlabamaMobileUSA

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