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**Von Neumann algebras and ergodic theory of group actions.**
*(English)*
Zbl 1178.46003

Oberwolfach Rep. 5, No. 4, 2763-2814 (2008). Abstracts from the workshop held October 26th–November 1st, 2008.

From the Introduction: The workshop Von Neumann Algebras and Ergodic Theory of Group Actions was organized by Dietmar Bisch (Vanderbilt University, Nashville), Damien Gaboriau (ENS Lyon), Vaughan Jones (UC Berkeley) and Sorin Popa (UC Los Angeles). It was held in Oberwolfach from October 26 to November 1, 2008. This workshop was the first Oberwolfach meeting on von Neumann algebras and orbit equivalence ergodic theory.

The first day of the workshop featured introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa’s deformation/rigidity techniques and applications to rigidity in II\(_1\) factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa).

A few of the highlights of the workshop were Vaes’ report on a new cocycle superrigidity result for non-singular actions of lattices in \(SL(n,\mathbb R)\) on \({\mathbb R}^n\) and on other homogeneous spaces (jointly with Popa), Ioana’s result showing that every sub-equivalence relation of the equivalence relation arising from the standard \(SL(2,\mathbb Z)\)-action on the 2-torus \({\mathbb T}^2\) is either hyperfinite or has relative property (T), and Epstein’s report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space.

Other talks discussed new results on fundamental groups of II\(_1\) factors, L\(^2\)-rigidity in von Neumann algebras, II\(_1\) factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra, and new results on topological rigidity and the Atiyah conjecture.

The first day of the workshop featured introductory talks to orbit equivalence and von Neumann algebras (Gaboriau), Popa’s deformation/rigidity techniques and applications to rigidity in II\(_1\) factors (Vaes), subfactors and planar algebras (Bisch), random matrices, free probability and subfactors (Shlyakhtenko), subfactor lattices and conformal field theory (Xu) and an open problem session (Popa).

A few of the highlights of the workshop were Vaes’ report on a new cocycle superrigidity result for non-singular actions of lattices in \(SL(n,\mathbb R)\) on \({\mathbb R}^n\) and on other homogeneous spaces (jointly with Popa), Ioana’s result showing that every sub-equivalence relation of the equivalence relation arising from the standard \(SL(2,\mathbb Z)\)-action on the 2-torus \({\mathbb T}^2\) is either hyperfinite or has relative property (T), and Epstein’s report on her result that every countable, non-amenable group admits continuum many non-orbit equivalent, free, measure preserving, ergodic actions on a standard probability space.

Other talks discussed new results on fundamental groups of II\(_1\) factors, L\(^2\)-rigidity in von Neumann algebras, II\(_1\) factors with at most one Cartan subalgebra, subfactors from Hadamard matrices, a new construction of subfactors from a planar algebra, and new results on topological rigidity and the Atiyah conjecture.