  We prove that a group $\Gamma$ admits a discrete topological (equivalently,
smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has
a Cayley complex embeddable -- with certain natural restrictions -- in one of
the following four 3-manifolds: (i) $\mathbb{S}^3$, (ii) $\mathbb{R}^3$, (iii)
$\mathbb{S}^2 \times \mathbb{R}$, (iv) the complement of a tame Cantor set in
$\mathbb{S}^3$.
