A combination theorem for special cube complexes | Annals of Mathematics

Abstract We prove that certain compact cube complexes have special finite covers. This means they have finite covers whose fundamental groups are quasiconvex subgroups of right-angled Artin groups. As a result we obtain linearity and the separability of quasiconvex subgroups for the groups we consider. Our result applies, in particular, to a compact negatively curved cube complex whose hyperplanes do not self-intersect. For a cube complex with word-hyperbolic fundamental group, we show that it is virtually special if and only if its hyperplane stabilizers are separable. In a final application, we show that the fundamental groups of every simple type uniform arithmetic hyperbolic manifolds are cubical and virtually special.

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数学年刊(Annals of Mathematics)

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