Appears in collection : 2014 - T1 - Random walks and asymptopic geometry of groups.

An invariant random subgroup (IRS) of a group is a random subgroup whose distribution is invariant under the conjugation action of the ambient group. IRS-es tend to behave like normal subgroups in the sense that results that hold for normal subgroups but not for arbitrary subgroups tend to generalize to IRS's. Also, weak convergence of IRS's translates to Benjamini-Schramm convergence of the corresponding quotient spaces. These phenomena can be exploited in various ways. In the talk I will survey the known results and directions and pose some questions.