Relative solidity for general biexact groups

Establish the relative solidity property for all biexact groups: given a biexact group Γ, a trace-preserving action Γ ⊳ N on a tracial von Neumann algebra N, and commuting von Neumann subalgebras P,Q ⊂ M := N ⋊ Γ, show that either P intertwines into N inside M (i.e., P ≺_M N) or Q is amenable relative to N inside M. This generalizes the known relative solidity results for hyperbolic groups and for weakly amenable biexact groups, and seeks to remove the weak amenability or abelian/W*CMAP assumptions on N under which partial results are known.

Background

The relative solidity property asserts that for certain classes of groups Γ acting on a tracial von Neumann algebra N, any pair of commuting subalgebras P,Q in the crossed product M = N ⋊ Γ must satisfy either P ≺_M N or Q is amenable relative to N. This property is known for hyperbolic groups and more generally for groups that are weakly amenable and biexact, stemming from Popa–Vaes’ relative strong solidity and related techniques.

Ozawa showed the property when N is abelian, Popa–Vaes established it for weakly amenable biexact groups, and Isozaki proved it when N has the W*CMAP. The paper highlights that extending this property to all biexact groups remains open, and formulates the precise conjecture seeking to establish it without additional assumptions on weak amenability or on N. The authors develop and prove a measure equivalence variant as a workaround in this work.