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For any torsion-free hyperbolic group Γ and any group  G  that is fully residually Γ, we construct algorithmically a finite collection of homomorphisms from  G  to groups obtained from Γ by extensions of centralizers, at least one of which is injective. When  G  is residually Γ, this gives a effective embedding of  G  into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to Γ.

Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers