Torsion-free acylindrically hyperbolic groups with trivial amenable radical satisfy the relative ISR-property, torsion-free hyperbolic groups satisfy the relative C*-ISR-property, and irreducible lattices like SL_d(Z) (d≥3) with trivial center satisfy an analogous property.
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Thompson's group F satisfies relative invariant subalgebra rigidity w.r.t. its commutator subgroup, with every [F,F]-invariant von Neumann subalgebra of L(F) equal to L(N) for normal N ⊴ F.
For every integer n at least 1, there exist i.c.c. groups G such that L(G) has precisely n G-invariant von Neumann subalgebras not arising from subgroups.
citing papers explorer
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On Relative Invariant Subalgebra Rigidity Property
Torsion-free acylindrically hyperbolic groups with trivial amenable radical satisfy the relative ISR-property, torsion-free hyperbolic groups satisfy the relative C*-ISR-property, and irreducible lattices like SL_d(Z) (d≥3) with trivial center satisfy an analogous property.
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Relative invariant subalgebra rigidity for Thompson's group $F$
Thompson's group F satisfies relative invariant subalgebra rigidity w.r.t. its commutator subgroup, with every [F,F]-invariant von Neumann subalgebra of L(F) equal to L(N) for normal N ⊴ F.
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Factors with prescribed number of invariant subalgebras not arising from subgroups
For every integer n at least 1, there exist i.c.c. groups G such that L(G) has precisely n G-invariant von Neumann subalgebras not arising from subgroups.