Relative solidity conjecture for biexact groups

Let Γ\Gamma be a biexact group acting trace-preservingly on a von Neumann algebra NN, and let

M:=NΓ.M:=N\rtimes\Gamma.

Let P,QMP,Q\subset M be commuting von Neumann subalgebras. Relative solidity conjecture. One has PMNP\prec_M N or QQ is amenable relative to NN inside MM. This property is known for hyperbolic groups and, more generally, for groups with the relative strong solidity property; whether it holds for all biexact groups is open and is a central difficulty in extending the unique prime factorization theorem to products of nonamenable biexact groups.

Sources & referencesView supporting material

Primary source

Changying Ding and Daniel Drimbe, “Relative solidity for biexact groups in measure equivalence”, arXiv:2503.24167 (2025).

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