On Relative Invariant Subalgebra Rigidity Property

TL;DR

This paper investigates the rigidity properties of certain groups, showing that torsion-free hyperbolic groups and some lattices have strong invariance characteristics in their von Neumann and C*-algebraic substructures.

Contribution

It establishes the relative ISR-property for torsion-free acylindrically hyperbolic groups and hyperbolic groups, and extends similar properties to higher-rank lattice groups.

Findings

Torsion-free acylindrically hyperbolic groups satisfy the relative ISR property.

Torsion-free hyperbolic groups satisfy the relative C*-ISR property.

Higher-rank lattices like SL_d(Z) have an analogous relative ISR property.

Abstract

A countable discrete group $\Gamma$ is said to have the relative ISR-property if for every non-trivial normal subgroup $N\trianglelefteq\Gamma$ and every von Neumann subalgebra $\mathcal{M}\subseteq L(\Gamma)$ invariant under conjugation by $N$, one has $\mathcal{M}=L(K)$ for some subgroup $K\le\Gamma$. Similarly, $\Gamma$ has the relative $C^*$-ISR-property if every $N$-invariant unital $C^*$-subalgebra $\mathcal{A} \subseteq C_r^*(\Gamma)$ is of the form $C_r^*(K)$. 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 $C^*$-ISR property. Furthermore, we establish an analogous relative ISR-property for irreducible lattices in higher-rank semisimple Lie groups, such as $\mathrm{SL}_d(\mathbb{Z})$ ($d \geq 3$), with trivial center.