Small cancellations over relatively hyperbolic groups and embedding theorems

Abstract

We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer to the well-known question of the existence of a finitely generated group $G$ other than $\mathbb Z/2\mathbb Z$ such that all nontrivial elements of $G$ are conjugate.