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Title: Cubulated relatively hyperbolic groups and virtual specialness.

Abstract: Ian Agol showed that hyperbolic groups acting geometrically on CAT(0) cube complexes are virtually special in the sense of Haglund鈥揥ise, the last step in the proof of the virtual Haken and virtual fibering conjectures. I will talk about a generalization of this result (also obtained independently by Groves and Manning), which states that cubulated relatively hyperbolic groups are virtually special provided the peripheral subgroups are virtually special in a way that is compatible with the cubulation. In particular, we deduce virtual specialness for cubulated groups that are hyperbolic relative to virtually abelian groups, extending Wise's results for limit groups and fundamental groups of cusped hyperbolic 3-manifolds. The main ingredient of the proof is a relative version of Wise's quasi-convex hierarchy theorem, obtained using recent results by Einstein, Groves, and Manning.

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