Lemma 97 ([Visible and independent evaluations]). After a further normal finite-index refinement of H₀ , the following conditions hold, and persist under further finite-index refinement. For each required nontrivial group M , there is …

Choose one nonidentity element in each required M . Its cyclic group is quasiconvex in D . For a specified pair, the two endpoint pairs in the boundary of D are disjoint: in a torsion-free hyperbolic group, cyclic subgroups sharing an endpoint have a common nontrivial power. Their positive and negative axes consequently have a bounded cancellation constant at a change from one generator to the other. Choose powers whose translation lengths exceed twice this bound and the fixed local-to-global threshold for quasigeodesics. The path for a reduced alternating word, after removing the bounded pieces at its joins, is a uniform local quasigeodesic, hence a global quasigeodesic. No such nonempty word is trivial, and the orbit map of the resulting rank-two free group is a quasi-isometric embedding. Thus these powers generate a quasiconvex free group, as required. Pass first to a compact special subgroup of finite index. Quasiconvex subgroups of a hyperbolic compact special group are virtual retracts . Apply this to suitable powers of each chosen cyclic group and each of the rank-two groups. A retraction onto a cyclic group gives its nonzero rational character; a retraction onto the rank-two group gives the two coordinate characters. Intersect the finitely many resulting subgroups and take a normal core in D . Restricting these characters still detects the required powers: a further power lying in the smaller subgroup has its former evaluation multiplied by a nonzero integer. The same observation preserves independence of the two evaluation functionals. Conjugation transports the resulting characters, proving the assertion for all conjugates. At a fixed normal finite-index stage only finitely many conjugacy tests on characters are needed, since inner conjugations by that subgroup act trivially on its characters.