Abstract:
The notion of measure equivalence of groups was introduced by Gromov as the measurable counterpart to the topological notion of quasi-isometry.  Another well-studied notion is that of $W^*\!$-equivalence which states that two groups $\Gamma$ and $\Lambda$ are $W^*\!$-equivalent if they have isomorphic group von Neumann algebras, i.e., $L\Gamma\eqsim  L\Lambda$. We introduce a coarser equivalence, which we call von Neumann equivalence, and show that it encapsulates both measure equivalence and $W^*\!$-equivalence. If time permits, we will also show that the new and wide class of groups, called properly proximal groups, introduced by Réemi Boutonnet, Adrian Ioana, and Jesse Peterson is also stable under von Neumann equivalence and thereby obtaining the first examples of non-inner-amenable, non-properly proximal groups. This is based on joint work with Jesse Peterson and Lauren Ruth.