  A matchbox manifold is a generalized lamination, which is a continuum whose
path components define the leaves of a foliation of the space. A matchbox
manifold is M-like if it has the shape of a fixed topological space M. When M
is a closed manifold, in a previous work, the authors have shown that if $\frak
M$ is a matchbox manifold which is M-like, then it is homeomorphic to a weak
solenoid. In this work, we associate to a weak solenoid a pro-group, whose
pro-isomorphism class is an invariant of the homeomorphism class of $\frak M$.
We then show that an M-like matchbox manifold is homeomorphic to a weak
solenoid whose base manifold has fundamental group which is non co-Hopfian;
that is, it admits a non-trivial self-embedding of finite index. We include a
collection of examples illustrating this conclusion.
