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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 .

Lemma 83 ([Unfilled intersections]). If B,C have exact relative embeddings into a row J , then for every fixed g∈J , the subgroup E=B∩gCg-1 is relatively hyperbolic and has exact relative embeddings into J , B , and gCg-1 .

Lemma 82 ([Unfilled comparison paths]). Fix two exact relative embeddings into a row, and fixed ordinary endpoint connectors. Expanded intrinsic geodesics have the following comparison. Large-angle apices occur on both paths in the same …

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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.

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