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We prove the bounded packing property for any abelian subgroup of a groupacting properly and cocompactly on a CAT(0) cube complex. A main ingredient ofthe proof is a cubical flat torus theorem. This ingredient is also used to showthat central HNN extensions of maximal free-abelian subgroups of compactspecial groups are virtually special, and to produce various examples of groupsthat are not cocompactly cubulated.

A Cubical Flat Torus Theorem and the Bounded Packing Property