  The a-adic numbers are those groups that arise as Hausdorff completions of
noncyclic subgroups of the rational numbers. We give a crossed product
construction of (stabilized) Cuntz-Li algebras coming from the a-adic numbers
and investigate the structure of the associated algebras. In particular, these
algebras are in many cases Kirchberg algebras in the UCT class. Moreover, we
prove an a-adic duality theorem, which links a Cuntz-Li algebra with a
corresponding dynamical system on the real numbers. The paper also contains an
appendix where a nonabelian version of the "subgroup of dual group theorem" is
given in the setting of coactions.
