  Let F be a finite subgroup of SL_2 (Z) (necessarily isomorphic to one of
Z/2Z, Z/3Z, Z/4Z, or Z/6Z), and let F act on the irrational rotational algebra
A_{\theta} via the restriction of the canonical action of SL_2 (Z). Then the
crossed product of A_{\theta} by F, and the fixed point algebra for the action
of F on A_{\theta}, are AF algebras. The same is true for the crossed product
and fixed point algebra of the flip action of Z/2Z on any simple d-dimensional
noncommutative torus A_{\Theta}. Along the way, we prove a number of general
results which should have useful applications in other situations.
