Maximal Haagerup subalgebras in L(Z2⋊SL2(Z))

We prove that L(SL2(k)) is a maximal Haagerup--von Neumann subalgebra in L(k2⋊SL2(k)) for k=Q and k=Z. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between L(SL2(k)) and L∞(Y)⋊SL2(k), where SL2(k)↷Y denotes the quotient of the algebraic action SL2(k)↷ˆk2 by modding out the relation ϕ∼ϕ′, where ϕ, ϕ′∈ˆk2 and ϕ′(x,y):=ϕ(−x,−y) for all (x,y)∈k2. As a by-product, we show L(PSL2(Q)) is a maximal von Neumann subalgebra in L∞(Y)⋊PSL2(Q); in particular, PSL2(Q)↷Y is a prime action.