On Relative Invariant Subalgebra Rigidity Property

A countable discrete group $\mathrm{\Gamma }$ is said to have the relative ISR-property if for every non-trivial normal subgroup $N⊴\mathrm{\Gamma }$ and every von Neumann subalgebra $\mathcal{M}\subseteq L(\mathrm{\Gamma })$ invariant under conjugation by $N$, one has $\mathcal{M}=L(K)$ for some subgroup $K\le \mathrm{\Gamma }$. Similarly, $\mathrm{\Gamma }$ has the relative $C^{\ast }$-ISR-property if every $N$-invariant unital $C^{\ast }$-subalgebra $\mathcal{A}\subseteq C_r^{\ast }(\mathrm{\Gamma })$ is of the form $C_r^{\ast }(K)$. We show that every torsion-free acylindrically hyperbolic group with trivial amenable radical satisfies the relative ISR property. Moreover, we also show that all torsion-free hyperbolic groups have the relative $C^{\ast }$-ISR property. Furthermore, we establish an analogous relative ISR-property for irreducible lattices in higher-rank semisimple Lie groups, such as ${\mathrm{S}\mathrm{L}}_d(\mathbb{Z})$ ($d\ge 3$), with trivial center.