Abstract: (Based on a joint work with Waltraud Lederle) A subgroup \Delta \in \Gamma in a countable group is called Boomerang, if for every \gamma \in \Gamma there is a sequence n_i such that \gamma^{n_i} \Delta \gamma^{-n_i} \rightarrow \Delta. With convergence in the Chabauty topology on the space of subgroups of \Gamma. In other words a boomerang is a recurrent point, with respect to the conjugation action of each element \gamma \in \Gamma separately. It is a direct consequence of Poincaré recurrence that every invariant random subgroup (IRS) of \Gamma is supported on boomerangs. 

Many results that are known to hold almost surely for IRSs can be proven deterministically for boomerangs. In particular I will discuss a version of Borel density. Then I will show that when \Gamma is a lattice in a simple Lie group of higher \mathbb{Q}-rank, then every boomerang in \Gamma is either of finite index or finite and central. This strengthens (and also gives a new proof for) the Nevo-Stuck-Zimmer theorem for such lattices.