Discrete group actions on 3-manifolds and embeddable Cayley complexes

We prove that a group $\mathrm{\Gamma }$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $\mathrm{\Gamma }$ 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 ${\mathbb{S}}^3$.