Hierarchies for Relatively Hyperbolic Virtually Compact Special Non-Positively Curved Cube Complexes

Abstract

Cube complexes and hierarchies of cube complexes have been studied extensively by Wise and feature prominently in Agol's proof of the Virtual Haken Conjecture for hyperbolic 3-manifolds. Among hyperbolic groups, Wise characterized hyperbolic virtually compact special groups as the hyperbolic groups that virtually admit a quasiconvex hierarchy terminating in finite groups. The main result of this thesis is that every relatively hyperbolic fundamental group of a virtually compact special non-positively curved cube complex virtually admits a quasiconvex hierarchy terminating in peripheral subgroups, answering a question due to Wise. The proof of the main theorem roughly follows the outline of Agol, Groves and Manning's New Proof of Wise's Malnormal Special Quotient Theorem for hyperbolic groups, but instead uses relatively hyperbolic geometric tools to prove that Wise's double dot hierarchy construction yields a quasiconvex hierarchy. Group theoretic relatively hyperbolic Dehn filling and the main theorem are used in the final chapter to provide a new proof of a relatively hyperbolic analog of Wise's malnormal special quotient theorem.