Invariant Random Subgroups in groups acting on rooted trees

An invariant random subgroup (IRS) of a countable group is a conjugation invariant probability distribution on the space of subgroups. There have been a number of recent papers studying IRS’s in various types of groups: lattices in Lie groups, the group of finitary permutations of a countable set, free groups, lamplighters and more.
In this talk we will consider IRS’s in groups acting of rooted trees, in particular the group of finitary automorphisms of a d-ary rooted tree. We exploit the action of these groups on the boundary of the tree to understand fixed point sets of ergodic IRS’s. We show that in the fixed point free case IRS’s behave like the ones in lattices in Lie groups, but if there are fixed points they resemble the ones in the finitary permutation group. Joint work with Ferenc Bencs.