  This paper uses Brin and Thickstun's theory of end reductions of non-compact
3-manifolds to study groups of covering translations of irreducible
contractible open 3-manifolds W which are not homeomorphic to R^3. We associate
to W an object S(W) called the simplicial complex of minimal R^2-irreducible
end reductions of W. Whenever W covers another 3-manifold the group of covering
translations is isomorphic to a fixed point free group of automorphisms of
S(W). We apply this result to give uncountably many examples of R^2-irreducible
such W which cover 3-manifolds with infinite cyclic fundamental group and
non-trivially cover only 3-manifolds with infinite cyclic fundamental group. We
also give uncountably many examples of R^2-irreducible W which are not
eventually end irreducible and do not non-trivially cover any 3-manifold.
