Deformations of II_1 factors with applications to their structural theory

Abstract

This work explores some aspects of the deformation/rigidity theory of II_1 factors. The work primarily addresses the deformation side of deformation/rigidity, developing techniques to accommodate natural deformations coming from geometry in order to study the structure of group and group-measure space factors. The results are divided into three chapters, each corresponding to an original research article.

The main result of the first chapter is that Popa's Cocycle Superrigidity Theorem for Bernoulli actions holds for the class of L^2-rigid groups defined by Peterson. This provides new examples of superrigid groups as well as a common treatment of the property (T) and product group cases. The second chapter extends a structural result, known as strong solidity, of Ozawa and Popa on group von Neumann algebras of discrete groups of motions of the hyperbolic plane to discrete groups of motions of n-dimensional hyperbolic space. The third chapter reworks Ozawa's theorem on the ``solidity' of group von Neumann algebras of Gromov hyperbolic groups from the perspective of deformation/rigidity theory. We are also able to obtain strong solidity for all i.c.c. Gromov hyperbolic groups and all i.c.c. lattices in connected, simple Lie groups of rank one.