Drilling hyperbolic groups

Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $⟨g⟩$, we define a {\it drilling of $G$ along $g$}, which is a relatively hyperbolic group pair $(\widehat{G},P)$. This is inspired by the well-studied procedure of drilling a hyperbolic $3$–manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with $2$-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair $(\widehat{G},P)$ has relatively hyperbolic boundary $S^2$. This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable.