Deformations of Margulis space-times with parabolics

Let $E$ be a flat Lorentzian space of signature $(2, 1)$. A Margulis space-time is a noncompact complete Lorentz flat $3$-manifold $E/\Gamma$ with a free isometry group $\Gamma$ of rank $g \geq 2$. We consider the case when $\Gamma$ contains a parabolic element. We show that sufficiently small deformations of $\Gamma$ still act properly on $E$. We use our previous work showing that $E/\Gamma$ can be compactified relative to a union of solid tori and some old idea of Carri\`ere in his famous work. We will show that the there is also a decomposition of $E/\Gamma$ by crooked planes that are disjoint and embedded in a generalized sense. These can be perturbed so that $E/\Gamma$ decomposes into cells. This partially affirms the conjecture of Charette-Drumm-Goldman.