  In this paper we construct odd finitely summable spectral triples based on
length functions of bounded doubling on noncommutative solenoids. Our spectral
triples induce a Leibniz Lip-norm on the state spaces of the noncommutative
solenoids, giving them the structure of Leibniz quantum compact metric spaces.
By applying methods of R. Floricel and A. Ghorbanpour, we also show that our
odd spectral triples on noncommutative solenoids can be considered as direct
limits of spectral triples on rotation algebras. In the final section we prove
a noncommutative Wiener's lemma and show that our odd spectral triples can be
defined to have an associated smooth dense subalgebra which is stable under the
holomorphic functional calculus, thus answering a question of B. Long and W.
Wu. The construction of the smooth subalgebra also extends to the case of
nilpotent discrete groups.
