We will associate two particular objects with a countable group $\Gamma$. Consider its subgroup space $\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. We can also associate with the group von Neumann algebra $L(\Gamma)$.

Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. They generalize the notion of normal subgroups. They strengthen the well-known Margulis’s normal subgroup Theorem, among many other remarkable results.

More recently, in a joint work with Hartman and Oppelmayer, we introduced the notion of Invariant Random Algebra (IRA), an invariant probability measure on the collection of subalgebras of $L(\Gamma)$.

Motivated by the works of Glasner and Lederle, in ongoing joint work with Yair Glasner, Yair Hartman, and Yongle Jiang, we introduce the notion of Boomerang subalgebras in the context of $L(\Gamma)$. In this talk, we shall connect these two very seemingly distant notions. If time permits, we shall show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We’ll also talk about its connection to understanding IRAs in such groups.