Finite groups acting on higher dimensional noncommutative tori

For the canonical action α   of SL2(Z)SL2(Z) on 2-dimensional simple rotation algebras AθAθ, it is known that if F   is a finite subgroup of SL2(Z)SL2(Z), the crossed products Aθ⋊αFAθ⋊αF are all AF algebras. In this paper we show that this is not the case for higher dimensional noncommutative tori. More precisely, we show that for each n≥3n≥3 there exist noncommutative simple ϕ(n)ϕ(n)-dimensional tori AΘAΘ which admit canonical action of ZnZn and for each odd n≥7n≥7 with 2ϕ(n)≥n+52ϕ(n)≥n+5 their crossed products AΘ⋊αZnAΘ⋊αZn are not AF (with nonzero K1K1-groups). It is also shown that the only possible canonical action by a finite group on a 3-dimensional simple torus is the flip action by Z2Z2. Besides, we discuss the canonical actions by finite groups Z5,Z8,Z10Z5,Z8,Z10, and Z12Z12 on the 4-dimensional torus of the form Aθ⊗AθAθ⊗Aθ.