  For every finitely generated abelian group G, we construct an irreducible
open 3-manifold $M_{G}$ whose end set is homeomorphic to a Cantor set and with
end homogeneity group of $M_{G}$ isomorphic to G. The end homogeneity group is
the group of self-homeomorphisms of the end set that extend to homeomorphisms
of the 3-manifold. The techniques involve computing the embedding homogeneity
groups of carefully constructed Antoine type Cantor sets made up of rigid
pieces. In addition, a generalization of an Antoine Cantor set using infinite
chains is needed to construct an example with integer homogeneity group.
Results about local genus of points in Cantor sets and about geometric index
are also used.
