  A weak solenoid is a foliated space defined as the inverse limit of finite
coverings of a closed compact manifold $M$. The monodromy of a weak solenoid
defines an equicontinuous minimal action on a Cantor space $X$ by the
fundamental group $G$ of $M$. The discriminant group of this action is an
obstruction to this action being homogeneous. The discriminant vanishes if the
group $G$ is abelian, but there are examples of actions of nilpotent groups for
which the discriminant is non-trivial. The action is said to be stable if the
discriminant group remains unchanged for the induced action on sufficiently
small clopen neighborhoods in $X$. If the discriminant group never stabilizes
as the diameter of the clopen set $U$ tends to zero, then we say that the
action is unstable, and the weak solenoid which defines it is said to be wild.
In this work, we show two main results in the course of our study of the
properties of the discriminant group for Cantor actions. First, the tail
equivalence class of the sequence of discriminant groups obtained for the
restricted action on a neighborhood basis system of a point in $X$ defines an
invariant of the return equivalence class of the action, called the asymptotic
discriminant, which is consequently an invariant of the homeomorphism class of
the weak solenoid. Second, we construct uncountable collections of wild
solenoids with pairwise distinct asymptotic discriminant invariants for a fixed
base manifold $M$, and hence fixed finitely-presented group $G$, which are thus
pairwise non-homeomorphic. The study in this work is the continuation of the
seminal works on homeomorphisms of weak solenoids by Rogers and Tollefson in
1971, and is dedicated to the memory of Jim Rogers.
