Hierarchies for relatively hyperbolic virtually special groups

Wise's quasiconvex hierarchy theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the virtual Haken conjecture.Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually compact special groups.We use this hierarchy to prove a generalization of Wise's malnormal special quotient theorem for relatively hyperbolic virtually compact special groups with arbitrary peripheral subgroups. 20F65, 20F67