Canonical representatives and equations in hyperbolic groups
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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]).
Supplementary Material (0)
- M. Gromov: Hyperbolic groups. Essays in group theory edited by S. Gersten, MSRI publication no. 8, Springer Berlin Heidelberg New York 1987, pp. 75–263
- I.G. Lysenok: On some algorithmic properties of hyperbolic groups. Math. USSR Izvestiya35 (1990) 145–163
- G.S. Makanin: Equations in a free group. Math. USSR Izvestiya21 (1983) 483–546
- A.A. Razborov: On systems of equations in a free group. Math. USSR Izvestiya25 (1985) 115–162
- Z. Sela: The isomorphism problem for hyperbolic groups I. Ann. Math. (to appear)
- Z. Sela: The isomorphism problem for hyperbolic groups II (In preparation)
- Z. Sela: Uniform embeddings of hyperbolic groups in Hilbert spaces. Israel J Math.80 (1992) 171–181
- Z. Sela and M. Ville: Bounded cohomology of small cancellation groups (in preparation).
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- Canonical representatives and equations in hyperbolic groups
Volume 120, Issue 1 , pp 489-512
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