Invariant random subgroups in combinatorics, dynamics and representation theory

Let G be a locally compact group. For example it could be a discrete group or a Lie group. A random closed subgroup of G, whose distribution is invariant under conjugation by elements of G is called an ”invariant random subgroup of G“ or IRS for short.

IRS turn out to be very useful in a surprisingly wide array of applications even outside of group theory. Yielding significant contributions to a-priori unrelated subjects such as these mentioned in the title.

I will survey some of these developments by stating one theorem in each of these subjects explaining exactly how IRS come into the picture.