  In this work, we investigate the dynamical and geometric properties of weak
solenoids, as part of the development of a "calculus of group chains"
associated to Cantor minimal actions. The study of the properties of group
chains was initiated in the works of McCord 1965 and Fokkink and Oversteegen
2002, to study the problem of determining which weak solenoids are homogeneous
continua. We develop an alternative condition for the homogeneity in terms of
the Ellis semigroup of the action, then investigate the relationship between
non-homogeneity of a weak solenoid and its discriminant invariant, which we
introduce in this work. A key part of our study is the construction of new
examples that illustrate various subtle properties of group chains that
correspond to geometric properties of non-homogeneous weak solenoids.
