Topology and Geometric Group Theory Seminar
The Cannon Conjecture states that a hyperbolic group with 2-sphere boundary is actually the fundamental group of a closed hyperbolic 3-manifold, and a relatively hyperbolic group with 2-sphere boundary is the fundamental group of some more general hyperbolic 3-manifold (for example finite volume). Filling and drilling in hyperbolic geometry give a way to go between these types of 3-manifolds. We show that even without assuming Cannon, these operations make some sense in the class of groups and group pairs with 2-sphere boundary. This is joint work with Groves, Haissinsky, Osajda, Sisto and Walsh.