The linear SL_2(Z)-action on T^n: ergodic and von Neumann algebraic aspects

The linear action of SL2(R) on Rn corresponding to its unique irreducible representation induces an action SL2(Z)↷Tn for every n⩾2 that factors through PSL2(Z) for n odd. Thus, setting Gn=SL2(Z) (respectively Gn=PSL2(Z)) for n even (respectively n odd), Gn↷Tn is free and ergodic, every ergodic sub-equivalence relation of the orbital equivalence relation is either amenable or rigid, and the fundamental group of the II1 factor Nn:=L∞(Tn)⋊Gn is trivial. For n even, L∞(Tn)⋊H is a maximal Haagerup subalgebra of Nn for every suitable maximal amenable subgroup H of SL2(Z).