Lemma 23. The fillings in Lemma 21 are virtually compact special.

Use the finite-index graph torus supplied there, and pass to the height power for which all its blocks are primitive and its periodic vertices are fixed. At every vertex of V make a separate copy in each incident block, retain one central vertex, and join the central vertex to its block copies by connector intervals. Collapsing each connector star recovers V , so this operation is a homotopy equivalence. The map acts on each block as before, sends central vertices according to p , and sends each connector linearly to its corresponding connector. The eventual image of this graph map is connected and contains all blocks, since each block map is primitive. Its remaining intervals and central vertices are periodic. Restriction to the eventual image does not change the mapping-torus group: inclusion and a sufficiently high power of the map satisfy Lemma 17. Connector orbits now give annuli joining the point tori and the block tori. This is a finite graph-of-groups decomposition over infinite cyclic groups. Each edge injection is detected by height, so is injective. The total group is hyperbolic, as a finite-index subgroup of J . Its cyclic edge groups are quasiconvex, and the finitely generated vertex groups are consequently quasiconvex. In particular every block torus is hyperbolic. Each block has a fixed vertex and satisfies the other hypotheses of Lemma 22, so is virtually compact special. The point tori are infinite cyclic. The quasiconvex hierarchy theorem now gives virtual compact specialness of the total group, and therefore of its finite-index supergroup J .