Title: Avatars of small cancellation
Abstract:
In general, given a finite presentation of a group, it is very difficult (in fact algorithmically impossible) to understand the group it defines. Small cancellation theory was developped as a combinatorial condition on a presentation that allows one to understand the group it represents. This very flexible construction has many applications to construct examples of groups with specific features.
Thurston's Dehn filling theorem is a fundamental theorem in 3-manifold theory asserting that given a finite volume hyperbolic manifold with a cusp, all but finitely many manifolds obtained by Dehn surgery on the cusp carry a hyperbolic metric.
I will show that these two results are two faces of a single theorem that applies to many groups acting on hyperbolic spaces