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 $g\ge 2$ 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 Carrière. 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.

arXiv comment14 pages. This was a part of the paper "Deformations of Margulis space-times with parabolics" (arXiv:2407.05932 [math.GT]). We separated the deformation part to this paper. We believe this is cleaner. The other paper will be devoted to quasi-disjointness of crooked planes, and will be made somewhat longer after a short time