  A matchbox manifold with one-dimensional leaves which has equicontinuous
holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a
classical Vietoris solenoid. In this work, we consider the problem, what can be
said about a matchbox manifold with equicontinuous holonomy dynamics, and all
of whose leaves have at most polynomial growth type? We show that such a space
must have a finite covering for which the global holonomy group of its
foliation is nilpotent. As a consequence, we show that if the growth type of
the leaves is polynomial of degree at most 3, then there exists a finite
covering which is homogeneous. If the growth type of the leaves is polynomial
of degree at least 4, then there are additional obstructions to homogeneity,
which arise from the structure of nilpotent groups.
