Theories of HNN-Extensions and Amalgamated Products

  • Markus Lohrey
  • Géraud Sénizergues
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4052)


It is shown that the existential theory of \({\mathbb G}\) with rational constraints, over an HNN-extension \({\mathbb G}=\langle {\mathbb H},t; t^{-1}at=\varphi(a) (a \in A) \rangle\) is decidable, provided that the same problem is decidable in the base group \({\mathbb H}\) and that A is a finite group. The positive theory of \({\mathbb G}\) is decidable, provided that the existential positive theory of \({\mathbb G}\) is decidable and that A and ϕ(A) are proper subgroups of the base group \({\mathbb H}\) with Aϕ(A) finite. Analogous results are also shown for amalgamated products. As a corollary, the positive theory and the existential theory with rational constraints of any finitely generated virtually-free group is decidable.


Free Product Graph Product Proper Subgroup Rational Constraint Existential Theory 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Markus Lohrey
    • 1
  • Géraud Sénizergues
    • 2
  1. 1.Universität Stuttgart, FMIGermany
  2. 2.Université Bordeaux I, LaBRIFrance

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