פעילויות השבוע

(A joint work with Waltraud Lederle) In order to avoid technicalities I will focus on one specific example for a higher $\mathbb{Q}$-rank lattice: the group $\Gamma = \mathrm{SL}_3(\mathbb{Z})$. This group exhibits strong rigidity properties, some of which are naturally expressed in topological terms. For example, one of the earliest rigidity results, the congruence subgroup property which was established independently by Mennicke and Bass-Milnor-Serre, can be expressed as an equality between two group topologies on $\Gamma$: The profinite and the congruence topologies. Margulis‘ celebrated normal subgroup theorem can be thought of as the statement that even the normal topology coincides with these two. Here the normal topology is defined by taking all infinite normal subgroups as a basis of identity neighborhoods for a topology on $\Gamma$. Together with Waltraud Lederle we introduce an a-priori much finer topology on $\Gamma$ called the boomerang topology and show that in fact even this topology coincides with the congruence topology. As a result we obtain a generalization of a rigidity theorem for probability measure preserving actions due to Nevo-Stuck-Zimmer.

In a joint work with Yair Hartman and Hanna Oppelmayer, we study the sub-von Neumann Algebras of the group von Neumann algebra $L\Gamma$. We will first show that $L\Gamma$ admits a maximal invariant amenable subalgebra. We will also introduce the notion of invariant probability measures on the space of sub-von Neumann algebras (IRAs) which is analogous to the concept of Invariant Random Subgroups. We shall show that amenable IRAs are supported on the maximal amenable invariant subalgebra.