A major theme in the study of von Neumann algebras is to investigate which structural aspects of the group extend to its von Neumann algebra. While this is a notoriously challenging problem, much progress in this direction has been made due to in large part to Popa’s Deformation/Rigidity theory. This approach analyzes von Neumann algebras which admit opposing properties “malleability” and “rigidity.” By exploiting the tension between these two properties, one can slowly unravel the structure of the algebra. I present recent progress made by Dan Hoff, Ben Hayes, Thomas Sinclair and myself in the case where the group has positive first L2-Betti number by further developing Popa’s framework
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