Cocycle superrigidity for profinite actions of irreducible lattices

[摘要] Let Γ\GammaΓ be an irreducible lattice in a product of two locally compact groups and assume that Γ\GammaΓ is densely embedded in a profinite group KKK. We give necessary conditions which imply that the left translation action Γ↷K\Gamma \curvearrowright KΓ↷K is “virtually” cocycle superrigid: any cocycle w ⁣:Γ×K→Δ{w\colon \Gamma\times K\rightarrow\Delta}w:Γ×K→Δ with values in a countable group Δ\DeltaΔ is cohomologous to a cocycle which factors through the map Γ×K→Γ×K0\Gamma\times K\rightarrow\Gamma\times K_0Γ×K→Γ×K0​ for some finite quotient group K0K_0K0​ of KKK. As a corollary, we deduce that any ergodic profinite action of Γ=SL2(Z[S−1])\Gamma=\mathrm{SL}_2(\mathbb Z[S^{-1}])Γ=SL2​(Z[S−1]) is virtually cocycle superrigid and virtually W∗^*∗-superrigid for any finite nonempty set of primes SSS.

[学科分类] 神经科学

[关键词] Cocycle superrigidity;deformation/rigidity theory;orbit equivalence;profinite action;irreducible lattice;strong ergodicity