Effective coherence of groups discriminated by a locally quasi-convex hyperbolic group

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We prove that every finitely generated group G discriminated by a locally quasi-convex torsion-free hyperbolic group Γ is effectively coherent: that is, presentations for finitely generated subgroups can be computed from the subgroup generators. We study G via its embedding into an iterated centralizer extension of Γ, and prove that this embedding can be computed. We also give algorithms to enumerate all finitely generated groups discriminated by Γ and to decide whether a given group, with decidable word problem, is discriminated by Γ. If Γ may have torsion, we prove that groups obtained from Γ by iterated amalgamated products with virtually abelian groups, over elementary subgroups, are effectively coherent.