Discrete group actions on 3-manifolds and embeddable Cayley complexes

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) $\mathbb {S}<^>2 \times \mathbb R$ , and (iv) the complement of a tame Cantor set in $\mathbb {S}<^>3$ . The fact that these are the only simply connected 3-manifolds that allow such actions is a consequence of the Thurston-Perelman geometrization theorem.