Rigidity for W*-McDuff groups

Juan Felipa Ariza Mejia; University of Iowa

The problem of understanding how much of the group G is remembered by the group von Neumann algebra L(G) has been a major research theme in the field of operator algebras. On one extreme of the rigidity question are W*-superrigid groups, those which are completely recoverable from the von Neumann algebra L(G). On the other extreme there is the class of icc amenable groups each of which generates the hyperfinite II1 factor. In between these two extremes of superrigidity, and complete lack thereof, there are many classes of non-amenable groups that display various rigidity phenomena. In our work, we introduce the first examples of groups whose lack of superrigidity can be completely characterized. Specifically, we introduce the notion of, and construct, groups that are McDuff W*-superrigid, that is groups G such that if L(G) = L(H) (for an arbitrary group H), then H = G x A for some icc amenable group A. We do this by introducing new geometric group theory methods to construct wreath-like product groups with a 2-cocycle with uniformly bounded support, and using the interplay between two types of deformations on their group von Neumann algebra to prove that such groups have infinite product rigidity. This is joint work with Ionut Chifan, Denis Osin and Bin Sun.