Tameness of Margulis space-times with parabolics

Let $\mathbf{E}$ be a flat Lorentzian space of signature $(2,1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $\mathbf{E}\mathrm{/}\mathrm{\Gamma }$ with a free holonomy group $\mathrm{\Gamma }$ of rank $$\mathbf{g}$, $\mathbf{g}$ \geq 2$\mathrm{.}Weconsiderthecasewhen$\Gamma$ contains a parabolic element. We obtain a characterization of proper $\mathrm{\Gamma }$-actions in terms of Margulis and Drumm-Charette invariants. We show that $\mathbf{E}\mathrm{/}\mathrm{\Gamma }$ is homeomorphic to the interior of a compact handlebody of genus $\mathbf{g}$ generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to estimate the translational parts of the affine transformation group and use some $3$-manifold topology.