Domains of discontinuity of Lorentzian affine group actions (Q6592123)

The authors study the question of non-emptyness of domains of proper discontinuity of discrete groups acting on affine spaces. The basic setup is as follows: Suppose that \(\Gamma < \mathbb{R}^n \rtimes O(n-1,1)< \mathrm{Aff}(\mathbb{R}^n)\) is a discrete subgroup such that the linear projection \(\ell: \Gamma\to O(n-1,1)\) is a faithful representation with convex-cocompact image. Given a representation \(\ell: \Gamma\to O(n-1,1)\), the affine action of \(\Gamma\) on \(\mathbb{R}^n\) is determined by a cocycle \(c\in Z^1(\Gamma, \mathbb{R}^{n-1,1}_\ell)\). The basic case \(n = 3\) was considered in the pioneering work by \textit{G. A. Margulis} [Sov. Math., Dokl. 28, 435--439 (1983; Zbl 0578.57012); translation from Dokl. Akad. Nauk SSSR 272, 785--788 (1983)]. The authors point out that even in this case non-emptyness of domains of discontinuity for arbitrary cocycle \(c\) was not known prior to the present work. The main result of the paper is:\N\NTheorem 2. Every subgroup \(\Gamma < \mathbb{R}^n \rtimes O(n-1,1)\) with faithful convex-cocompact linear representation \(\ell: \Gamma\to O(n-1,1)\) acts properly discontinuously on a nonempty open subset of the Lorentzian space \(\mathbb{R}^{n-1,1}\).\N\NThe proof of the theorem is based on the results and methods developed by the first author et al. [Geom. Topol. 22, No. 1, 157--234 (2018; Zbl 1381.53090)].