New examples of W* and C* superrigid groups, strong rigidity for graph product groups, and cocycle superrigidity for joinings

The von Neumann algebra was originally introduced by John von Neumann, motivated by its formulations of quantum mechanics and representation theory. His works along with Murray provided constructions which transform well-understood mathematical objects (in our case we study groups and actions of groups on probability spaces by groups) into von Neumann algebras. Since their inception, there has been various attempts to characterize these objects by exploiting the structure of the generating objects.

If equivalent von Neumann algebras arise from different groups, the first is called the source and the second is called the target, we aim to understand under what assumptions on the source group, is the target group necessarily indistinguishable. Similarly, in the dynamic setting one can ask under what assumptions on the source action and group is the target the same. In this thesis we explore different classes of groups and actions that are completely recognizable from their associated von Neumann algebra.