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