Dehn surgery is a classical operation in which one converts one three-manifold to another by first removing a solid torus, and then gluing it back in in a different way. The first operation is called “drilling” and the second “filling”. Both of these operations have group-theoretic interpretations in the world of hyperbolic and relatively hyperbolic groups. I will explain those interpretations and applications related to the Cannon conjecture (a special case of Wall’s conjecture about $PD(n)$ groups).

The most recent work is joint with Groves, Haïssinsky, Osajda, Sisto, and Walsh.