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 Carrière 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.