Specializing cubulated relatively hyperbolic groups

TL;DR

This paper extends the concept of virtual specialness from hyperbolic groups to relatively hyperbolic groups that are cocompactly cubulated, using recubulation and Dehn filling techniques, and applies it to the Relative Cannon Conjecture.

Contribution

It generalizes Agol's result to a broader class of relatively hyperbolic groups with minimal assumptions on parabolic subgroups.

Findings

Cocombing of relatively hyperbolic groups to virtually special groups

Application to the Relative Cannon Conjecture

Recovery of a theorem by Oregón-Reyes using new methods

Abstract

In arXiv:1204.2810 Agol proved the Virtual Haken and Virtual Fibering Conjectures by confirming a conjecture of Wise: Every cubulated hyperbolic group is virtually special. We extend this result to cocompactly cubulated relatively hyperbolic groups with minimal assumptions on the parabolic subgroups. Our proof proceeds by first recubulating to obtain an improper action with controlled stabilizers (a weakly relatively geometric action), and then Dehn filling to obtain many cubulated hyperbolic quotients. We apply our results to prove the Relative Cannon Conjecture for certain cubulated or partially cubulated relatively hyperbolic groups. One of our main results (Theorem A) recovers via different methods a theorem of Oreg\'on-Reyes (arXiv:2003.12702).