The notion of measure equivalence of discrete groups has been introduced by Gromov as a measurable variant of the topological notion of quasi-isometry. Measure equivalence of groups is tightly related to the theory of II_1 factors: if G and H are measure equivalent, then they admit free ergodic probability measure preserving action for which their von Neumann algebras are stably isomorphic. Also, two groups G and H are called W*-equivalent if their group von Neumann algebras are stably isomorphic.

A few years ago, it was discovered that there is an even coarser notion of equivalence of groups, coined von Neumann equivalence, which is implied by both measure equivalence and W*-equivalence. In this talk I will present a unique prime factorization for products of hyperbolic groups up to von Neumann equivalence. This is joint work with Stefaan Vaes.