Rigidity results for von Neumann algebras arising from mixing extensions of profinite actions of groups on probability spaces

Invariant subalgebras of von Neumann algebras arising from negatively curved groups

“…Our main interest in the study of arrays and Gaussian dilation lies in understanding the structure of Γ-invariant subalgebras of L(Γ). In [CD19] Gaussian deformations were used to understand the structure of invariant subfactors of the group von Neumann algebras associated with "negatively curved groups".…”

“…A natural analogue of the aforementioned Galois correspondence results in the setting of discrete groups is to study the structure of invariant von Neumann subalgebras of the associated group von Neumann algebra. This study was undertaken by two of the authors of this paper in [CD19], where the structure of Γ-invariant factors inside L(Γ), for Γ an icc discrete group, was investigated. In particular, it was established that [CD19, Theorem 3.15] for any Γ-invariant II 1 factor N ⊂ L(Γ) there exists a normal subgroup Λ ⊳ Γ, with N ⊆ L(Λ) ⊆ N ∨ N ′ ∩ L(Γ).…”

“…Motivated by the results in [Pe14, AB21, CD19], Kalantar and Panagopoulos [KP21] showed that any Γ-invariant von Neumann subalgebras of L(Γ) arise as the group von Neumann algebra of a normal subgroup, with Γ being an irreducible lattice in a connected semi-simple Lie group G with trivial center, no nontrivial compact factors, and such that all simple factors of G are higher rank. This result as well as [CD19] further inspired Amrutam and Jiang to study invariant von Neumann subalgebras of L(Γ) for other classes of groups complementary to lattices in higher rank groups [AJ22]. Specifically they established the ISR property for L(Γ), whenever Γ is a nonamenable torsion free group that is either hyperbolic, or has positive first L 2 -Betti number and satisfies Peterson-Thom's condition ( * ) in [PT11].…”

“…We notice that Theorem B applies to a vast category of geometric groups including all icc hyperbolic groups, all icc relative hyperbolic groups, most mapping class groups, all outer automorphism groups of free groups with at least three generators, all nontrivial graph product groups whose underlying graph does not admit a visual splitting, etc. To this end, we emphasize that our approach for Theorem B is very different in essence from the deformation/rigidity theory methods employed in [CD19], using more generic von Neumann algebraic technique (Theorems 3.1 and 4.1) in combination to very strong group theoretic properties of the hyperbolically embedded subgroups of these groups (Theorem 2.7).…”

“…In the second part of the paper, we shift our perspective and study Γ-invariant subalgebras of L(Γ) via methods similar to the ones used in [CD19]. As in that case, the negative curvature information we heavily exploit, is the existence of unbounded (quasi)cocycles on Γ that are valued into its left regular representation.…”