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Title: Classification and Rigidity for von Neumann Algebras Arising from Property (T) Groups.

Abstract: In the 1930s, F. Murray and J. von Neumann discovered a natural way to associate a von Neumann algebra L(G) with every countable group G. Classifying L(G) in terms of  G has emerged as a natural yet challenging problem, as these algebras tend to have very limited information about the underlying group. This limitation is exemplified by A. Connes’ celebrated result (1976), which asserts that all amenable groups with infinite nontrivial conjugacy classes (icc) yield isomorphic von Neumann algebras; thus, aside from amenability, L(G) does not retain any information about G.

In contrast, the classification of non-amenable groups remains wide open and significantly more complex. Instances where the von Neumann algebraic structure is sensitive to various algebraic properties of groups have been identified through Popa’s deformation/rigidity theory. Notably, a famous conjecture by A. Connes (1982) predicts that all icc property (T) groups G are completely recognizable from L(G). In my talk, I will introduce the first examples of property (T) groups that satisfy this conjecture. These groups, termed wreath-like products, arise naturally in the context of group-theoretic Dehn filling.

Wreath-like product groups can also be employed to compute the outer automorphism groups and the fundamental groups of property (T) von Neumann algebras, addressing other well-known open problems posed by A. Connes (1994), V.F.R. Jones (2000), and S. Popa (2006). Finally, we will discuss a natural generalization of A. Connes’ conjecture for non-icc groups, along with some recent evidence supporting it.

This work is based on several recent collaborations with A. Ioana, A. Fernandez Quero, D. Osin, B. Sun, and H. Tan.