Tattwamasi Amrutam
Ben Gurion University of the Negev
August 12, 2024
Boomerang subalgebras of the group von Neumann algebra: Consider a countable discrete group $\Gamma$ and its subgroup space-$\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. $\text{Sub}(\Gamma)$ is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on $\{0,1\}^{\Gamma}$. The normal subgroups of $\Gamma$ are the fixed points of $(\text{Sub}(\Gamma), \Gamma)$. Furthermore, the $\Gamma$-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs). Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. Among many other remarkable results, they strengthen the well-known Nevo-Stuck-Zimmer-Theorem. To a countable discrete group $\Gamma$, we can also associate an algebraic object $L(\Gamma)$, called the group von Neumann algebra. 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 an 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 show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.
Ben Gurion University of the Negev
August 12, 2024
Boomerang subalgebras of the group von Neumann algebra: Consider a countable discrete group $\Gamma$ and its subgroup space-$\text{Sub}(\Gamma)$, the collection of all subgroups of $\Gamma$. $\text{Sub}(\Gamma)$ is a compact metrizable space with respect to the Chabauty topology (the topology induced from the product topology on $\{0,1\}^{\Gamma}$. The normal subgroups of $\Gamma$ are the fixed points of $(\text{Sub}(\Gamma), \Gamma)$. Furthermore, the $\Gamma$-invariant probability measures of this dynamical system are known as invariant random subgroups (IRSs). Recently, Glasner and Lederle have introduced the notion of Boomerang subgroups. Among many other remarkable results, they strengthen the well-known Nevo-Stuck-Zimmer-Theorem. To a countable discrete group $\Gamma$, we can also associate an algebraic object $L(\Gamma)$, called the group von Neumann algebra. 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 an 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 show that every Boomerang subalgebra of a torsion-free non-elementary hyperbolic group comes from a Boomerang subgroup. We shall also discuss its connection to understanding IRAs in such groups.