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

We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $\Gamma$ 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 $\Gamma$, and prove that this embedding can be computed. We also give algorithms to enumerate all finitely generated groups discriminated by $\Gamma$ and to decide whether a given group, with decidable word problem, is discriminated by $\Gamma$. If $\Gamma$ may have torsion, we prove that groups obtained from $\Gamma$ by iterated amalgamated products with virtually abelian groups, over elementary subgroups, are effectively coherent.