In previous work, together with Yair Glasner, I introduced boomerang subgroups of countable groups.

We prove that in many lattices, like in $SL_n(\mathbb{Z})$, every boomerang subgroup is finite and central or has finite index; generalizing the Margulis Normal Subgroup theorem and providing a deterministic generalization of the Stuck-Zimmer rigidity theorem for these lattices.
But few examples of boomerang subgroups are known and it seems not easy to construct them. In this talk, I will give not a concrete, but a generic construction using measurable full groups and conservative actions.

Based on joint work with Yair Glasner and Tobias Hartnick.