Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

Abstract

Abstract. We study equivalence relations that arise from translation actions Γ G which are associated to dense embeddings Γ < G of countable groups into second countable locally compact groups. Assuming that G is simply connected and the action Γ G is strongly ergodic, we prove that Γ G is orbit equivalent to another such translation action Λ H if and only if there exists an isomorphism δ : G → H such that δ(Γ) = Λ. If G is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of Γ on homogeneous spaces of the form G/Σ, where Σ < G is either a discrete or an algebraic subgroup. We also prove that if G is simply connected and the action Γ G has property (T), then any cocycle w : Γ × G → Λ with values into a countable group Λ is cohomologous to a homomorphism δ : Γ → Λ. As a consequence, we deduce that the action Γ G is orbit equivalent superrigid: any free nonsingular action Λ Y which is orbit equivalent to Γ G, is necessarily conjugate to an induction of Γ G.

Local spectral gap in simple Lie groups and applications

“…For translation actions on compact groups, strong ergodicity is implied by the spectral gap property, which is now known to hold in considerably large generality by [BG06,BG10,BdS14]. On the other hand, in the case of translation actions on locally compact non-compact groups, strong ergodicity seems much harder to work with, and so far could only be checked in two rather specific situations (see [Io14,Propositions G and H]).…”

Rigidity for Von Neumann Algebras

“…Assuming that Γ (G, m G ) has spectral gap, it is shown in [Io13] that the actions Γ G and Λ H are orbit equivalent iff they are conjugate. Subsequently, this has been generalized in [Io14] to the case when G and H are arbitrary, not necessarily compact, connected Lie groups with trivial centers. The only difference is that, in the locally compact setting, the spectral gap assumption no longer makes sense and has to be replaced with the assumption that the action Γ (G, m G ) is strongly ergodic.…”

A remark on fullness of some group measure space von Neumann algebras

“…Lemma 10 (cf. [Ioa14,Proposition G]). Let G be a second countable locally compact group, Γ G be a lattice, and H G be a closed subgroup such that the action Γ\G H is strongly ergodic.…”