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In Hyperbolic groups (page 82), Gromov claims that, by a standard application of Thurston's method of geodesic (hyperbolic) simplices, it can be prove that a hyperbolic group contains finitely many pairwise non conjugate subgroups isomorphic to a fixed one-ended finitely presented group.

I took a look on the usual references dealing with hyperbolic groups, but I didn't find any details. Do you know references about this result and/or Thurston's method?

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There is a proof of this theorem due to Thomas Delzant:

T. Delzant, L’image d’un groupe dans un groupe hyperbolique, Comment. Math. Helv. 70 (1995), no. 2, 267–284.

There is also a version for relatively hyperbolic groups du to Francois Dahmani:

Accidental Parabolics and Relatively Hyperbolic Groups, Israel Journal of Mathematics, Volume 153, Issue 1, pp 93–127

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    $\begingroup$ There is also a different argument using the Rips machine and the shortening argument that shows that there are only finitely many monomorphisms (up to precomposition with an automorphism) from a finitely generated one-ended group to a fixed hyperbolic group. In the torsion-free case I think you can find it in the paper "Structure and rigidity ... " by Rips and Sela. The case with torsion needs a version of the Rips machine due to Guirardel. $\endgroup$ – Richard Weidmann Mar 12 '17 at 19:22
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    $\begingroup$ PS: "Structure and rigidity ... " needs finitely presented, but the argument goes through using the version of the Rips machine due to Sela in "Acylindrical accessibilty" which was then generalized (and fixed) by Guirardel. $\endgroup$ – Richard Weidmann Mar 12 '17 at 19:44

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