140 INDEX OF NOTATION AND DEFINED TERMS deep, 58 degenerate, 89 Dehn complex, 9 diagram in a complex X, 15 disk diagram, 15 disk diagram D in X∗, 79 distinct conjugates, 25 doubly-external, 83 dual, 13, 53 dual CAT(0) cube complex, 2 dual curves, 16 elevation, 41 expanded edge group, 127 extends, 71 external, 83 faces, 7 fiber-product, 35 flag complex, 8 full, 29, 117 full subcomplex, 22 Fully residually free groups, 132 geodesic, 14 graph group, 12 halfspace carriers, 24 halfspaces, 53 height, 25 hexagon move, 17 hierarchy, 2, 131 honeycomb, 64 horizontal, 11 hyperplane, 13 immersed hyperplane, 31 immersed walls, 72 immersion, 10 in parallel, 89 incompressible surfaces, 131 innerpath, 82 internal, 83 internal part, 89 interosculate, 32 intersecting conjugator, 26, 122 ladder, 84 length, 121 limit groups, 132 linear separation, 65 link, 8 local isometry, 22 Magnus subgroup, 129 Magnus-Moldavanskii hierarchy, 129 malnormal, 25 malnormal quasiconvex hierarchy, 107 map of cubical presentations, 118 metric small-cancellation, 77 midcells, 57 midcube, 13 missing corners of squares, 23 monkey-tails, 91 multiply-external, 83 no missing shells, 117, 119 nonpositively curved, 8 nontrivial, 82 one-relator group, 129 open cubical neighborhood, 44 open halfspaces, 53 orientation, 53 outerpath, 21, 82 parallel, 40 piece, 77, 82 pieces, 86, 89 prime, 9 profinite topology, 34 properly, 65 quasiconvex, 25 quasiconvex hierarchy, 121 quasiflats, 65 raag, 12 rectangles, 86 reduced, 84, 86, 101 relatively malnormal, 26 removed, 85 Removing bigons, 86 Replacing cone-cells, 86 residually finite, 34 right-angled Artin group, 12 self-osculates, 32 separable, 34 separating, 53 shards, 89 shell, 80 singly-external, 83 size, 86 sparse, 65 special, 31 spur, 80, 82 strongly convex, 104 superconvex, 29
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