Boundaries of Dehn fillings

Abstract

We begin an investigation into the behavior of Bowditch and Gromov boundaries under the operation of Dehn filling. In particular, we show many Dehn fillings of a toral relatively hyperbolic group with $2$–sphere boundary are hyperbolic with $2$–sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture.