Rigidity for equivalence relations on homogeneous spaces

TL;DR

This paper investigates the rigidity properties of equivalence relations generated by lattice actions on homogeneous spaces, establishing conditions under which these relations and associated von Neumann algebras exhibit rigidity or hyperfiniteness.

Contribution

It proves the rigidity of actions of lattices in semisimple Lie groups on homogeneous spaces and characterizes subequivalence relations under certain conditions.

Findings

Actions of lattices in semisimple Lie groups on homogeneous spaces are rigid.

Von Neumann algebras from these actions have property (T) if the group has property (T).

Subequivalence relations are either hyperfinite or rigid when the adjoint action is amenable.

Abstract

We study Popa's notion of rigidity for equivalence relations induced by actions on homogeneous spaces. For any lattices $\Gamma,\Lambda$ in a semisimple Lie group $G$ with finite center and no compact factors we prove that the action $\Gamma\curvearrowright G/\Lambda$ is rigid. If in addition $G$ has property (T) then we derive that the von Neumann algebra $L^{\infty}(G/\Lambda)\rtimes\Gamma$ has property (T). We also show that if the adjoint action of $G$ on the Lie algebra of $G$ - $\{0\}$ is amenable (e.g. if $G=SL_2(\Bbb R)$), then any ergodic subequivalence relation of the orbit equivalence relation of the action $\Gamma\curvearrowright G/\Lambda$ is either hyperfinite or rigid.