4 1. OVERVIEW here) that can be obtained by cutting along hyperplanes in a finite cover ̂. Thus ¯′ is virtually special by Theorem 1.4. Theorem 1.7 (Quasiconvex Hierarchy). Suppose G is hyperbolic and has a quasiconvex hierarchy. Then G is virtually compact special. Proving Theorem 1.7 depends on proving virtual specialness of the amal- gamated free products and HNN extensions that arise at each stage of the hierarchy. Given a splitting, say G = A∗Ct=D, the plan is to find a finite index subgroup G′ with an almost malnormal quasiconvex hierarchy and conclude by applying Theorem 1.4. To do this, we verify separability of C by apply- ing Theorem 1.6 to quotient subgroups of C using an argument inducting on Height(G,C). This idea of repeatedly quotienting with an induction on height was independently discovered by Agol-Groves-Manning. Generalizations of Theorem 1.7 hold in many (and conjecturally all) cases when G is hyperbolic relative to abelian subgroups. We describe how to deduce these generalizations from Theorem 1.7. The proof of separabil- ity essentially involves a generalization of Theorem 1.7 to provide virtually special parabolic quotienting. However cubulating requires some additional work. 1.1. Applications We describe three notable classes of groups with quasiconvex hierarchies in Chapter 16: Limit groups have hierarchies given by Kharlamopovich-Miasnikov and by Sela, and are thus virtually special. Every one-relator group has a Magnus-Moldavanskii hierarchy and for one relator groups with torsion this hierarchy is a quasiconvex hierarchy. (Though technically one must pass to a torsion-free finite index subgroup.) This resolves Baumslag’s conjecture that every one-relator group with tor- sion is residually finite indeed they are virtually special and thus linear and have separable quasiconvex subgroups. For a hyperbolic 3-manifold M with an incompressible surface S the Haken hierarchy of M yields a quasiconvex hierarchy for π1M provided π1S is geometrically finite, and so π1M is virtually special. When the hyper- bolic 3-manifold has a geometrically finite incompressible surface, we thus find that π1M is subgroup separable: Indeed, the geometrically finite sub- groups are quasiconvex and hence separable using virtual specialness, and the virtual fiber subgroups are easily seen to be separable, and there are no other subgroups by the Tameness Theorem [1, 13]. A second corollary is that when the hyperbolic 3-manifold M is Haken, in the sense that it has an incompressible surface S, then M is virtually fibered. Indeed, either S is a virtual fiber, or it is geometrically finite, and in the latter case π1M is virtually in a raag and thus virtually RFRS and so Agol’s fibering criterion applies [2].
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