Drilling hyperbolic groups

Abstract

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 $(\hat{G},P)$. 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 $(\hat{G},P)$ 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.

83 pages, 2 figures. v2: Introduction rewritten, various other small corrections. Note that the letter-labelled main results have been rearranged and therefore changed labels