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Canonical representatives and equations in hyperbolic groups

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Summary

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]).

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References

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Authors and Affiliations

  1. Department of Mathematics, The Hebrew University, 91904, Jerusalem, Israel

    E. Rips & Z. Sela

  2. Columbia University, 10027, New York, NY, USA

    Z. Sela

Authors
  1. E. Rips
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  2. Z. Sela
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Additional information

Oblatum 1-1992 & 1-XI-1994

Partially supported by NSF grant DMS-9305848

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Rips, E., Sela, Z. Canonical representatives and equations in hyperbolic groups. Invent Math 120, 489–512 (1995). https://doi.org/10.1007/BF01241140

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  • DOI: https://doi.org/10.1007/BF01241140

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