Lemma 98 ([Separating actual cyclic pairs]). There is a normal finite-index subgroup H D , contained in the prescribed H₀ , such that every specified pair ⟨a⟩,⟨b⟩ has nontrivial images with trivial intersection in D/H . Every specified …

List the conjugacy types of the maximal cyclic groups containing the specified generators. In a torsion-free hyperbolic group these form a malnormal quasiconvex family. Apply the malnormal special quotient theorem with sufficiently deep power kernels in these maximal cyclic groups, retaining its designated finite-index normal subgroups P_i P_i , and also apply the relative Dehn filling theorem . We obtain a hyperbolic virtually compact special quotient D' in which the chosen maximal cyclic groups have the exact prescribed finite orders. For P_i=⟨c_i⟩ , choose its power exponent divisible by both [P_i: P_i] and [P_i:P_i∩H_0] , as well as by all prescribed order factors. Increase these multiples to meet the finite avoidance and lower-bound requirements. The power kernels then lie in both P_i and H_0 , as required by the two quotient constructions. Thus if a=c^k , with c a maximal cyclic generator, choosing the order of c divisible by |k|r makes the order of a divisible by r . We verify separation in D' , including when the two marked maximal cyclic groups are conjugate. Use the coned relative Cayley graph, with its uniform hyperbolicity constant and the uniform original-peripheral angle dictionaries of Section. The constants come from fixed relative relators and the uniform linear relative isoperimetric bound ; hence they are chosen before the power exponents. Preserve the distinctness of the finitely many marked apex cosets by the finite peripheral membership tests. Let v≠w be two resulting marked apices, and let I be the intersection of their stabilizers. For u∈I , compare a geodesic γ from v to w with uγ . Thin bigons bound the angle between their first germs by a constant independent of u and of the filling exponents. For a short geodesic use the path through w ; otherwise join the two geodesics a fixed distance from v . This gives an avoiding path of uniformly bounded length. Normalize v to the apex of a maximal cyclic group P=⟨c⟩ . Its first ordinary neighbor has coordinate x∈ P , so the first-neighbor displacement for uγ is x^ -1 ux=u , since P is cyclic. The uniform angle dictionary therefore represents every u∈I by one of a fixed finite list of powers c^e , say |e|≤B . If | P|=n>3B , no nontrivial subgroup of P is contained in these residues: every nontrivial subgroup of a cyclic group of order n has an element at cyclic distance at least n/3 from the identity. Thus I=1 . This proves the required pairwise separation in D' . Finally use residual finiteness of D' . For each of the finitely many marked finite subgroup pairs A,B≤D' , separate from the identity every nonidentity element xy^ -1 , x∈A, y∈B ; also separate the nonidentity elements of each marked finite subgroup. The resulting finite quotient preserves both the exact orders and the trivial pairwise intersections. Take its product with the existing quotient D/H_0 , which factors through D' because the power-filling kernel lies in H_0 . The kernel of the resulting map from D is the required H .