The stability of Margulis space-times with parabolic holonomy elements

Abstract

Let $E$ be a flat Lorentzian space of signature $(2,1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $E/\Gamma$, where the holonomy group $\Gamma$ is a free group of rank $g\geq 2$ acting freely and properly discontinuously by isometries. We consider the case where $\Gamma$ contains a parabolic element. We show that sufficiently small deformations of $\Gamma$ still act properly discontinuously on $E$ provided their linear parts are Fuchsian; moreover, the number of conjugacy classes of parabolic elements may increase or decrease under deformation. Our proof combines our previous compactification of $E/\Gamma$ relative to parabolic holonomy elements with a partial generalization of the work of Carri\`ere. However, this result depends only on the parts on parabolic actions of our earlier work. We believe that the shortness of the proof of this openness result is of independent interest.