Inventiones mathematicae

, Volume 120, Issue 1, pp 489–512

Canonical representatives and equations in hyperbolic groups

  • E. Rips
  • Z. Sela

DOI: 10.1007/BF01241140

Cite this article as:
Rips, E. & Sela, Z. Invent Math (1995) 120: 489. doi:10.1007/BF01241140


We use canonical representatives in hyperbolic groups to reduce the theory of equations in (torsion-free) hyperbolic groups to the theory in free groups. As a result we get an effective procedure to decide if a system of equations in such groups has a solution. For free groups, this question was solved by Makanin [Ma]|and Razborov [Ra]. The case of quadratic equations in hyperbolic groups has already been solved by Lysenok [Ly]. Our whole construction plays an essential role in the solution of the isomorphism problem for (torsion-free) hyperbolic groups ([Se1],[Se2]).

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • E. Rips
    • 1
  • Z. Sela
    • 1
    • 2
  1. 1.Department of MathematicsThe Hebrew UniversityJerusalemIsrael
  2. 2.Columbia UniversityNew YorkUSA

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