Corollary 84 ([Unfilled nonparabolic double cosets]). For fixed exact relative embeddings of B,C in a row J , the double cosets BgC having nonparabolic intersection B∩gCg^-1 have representatives in a fixed finite subset of J .

A nontrivial nonparabolic subgroup of a torsion-free relatively hyperbolic group contains a loxodromic element. Compare long paths in the two embedded groups tending to its two endpoints. The central comparison in Lemma 82 ([Unfilled comparison paths]) gives paired frame points b∈B and gc∈gC with b^ -1 gc in a fixed finite list. This is a representative of BgC . The list is independent of g , because only central comparison constants are used, not the remote endpoint connectors.