Profinite properties of groups

A residually finite group G is absolutely profinitely rigid if the isomorphism type of its profinite completion determines the isomorphism type of G. Absolute profinite rigidity is known for only a handful of groups. A related problem is to determine the properties of groups which are preserved under the isomorphism of its profinite completion. In this talk we will aim to give an overview of the state of the art of the subject and, time-permitting, discuss new results on profinite properties of free-by-cyclic groups. This is joint work with Sam Hughes.