A stability theorem for non-abelian actions on threadlike homogeneous spaces (Q1625126)

Given a space \(M\) with an action of a connected Lie group \(G\), a fundamental problem is the study of the geometry and dynamics of subgroups of \(G\) acting properly discontinuously on \(M\). The problem of describing deformations was first advocated by T. Kobayashi in 1993 for the general non-Riemannian case. Let \(H\) be a closed connected subgroup of \(G\) and \(\Gamma \) a discrete subgroup of \(G\) which acts freely and properly on \(M = G/H\). The parameter space of the action of \(\Gamma\) on \(G/H\) is defined by: \[ \mathcal R (G, \Gamma, H):= \Biggl\{ \varphi \in \mathrm{Hom}(\Gamma, G) \biggl{|} \begin{matrix} \varphi\text{ is injective}, \varphi(\Gamma) \text{ discrete and} \\ \text{acts properly and freely on } G/H\\ \end{matrix} \Biggl\}. \] A homomorphism \(\varphi \in\mathcal R(G, \Gamma, H)\) is said to be stable, if there is an open set in \(\mathrm{Hom}(\Gamma, G)\) which contains \(\varphi\) and is contained in \(\mathcal R(G, \Gamma, H)\). As an application of the general theory, \textit{T. Kobayashi} and \textit{S. Nasrin} [Int. J. Math. 17, No. 10, 1175--1193 (2006; Zbl 1124.57015)] studied the setup of a properly discontinuous action of a discrete subgroup \(\Gamma\) on \(\mathbb R^{k+1} \simeq G/H\) through a certain two-step nilpotent affine transformation group \(G\) of dimension \(2k+1\) when the connected subgroup \(H\) is \(\mathbb R^k\). \textit{A. Baklouti} et al. [Int. J. Math. 22, No. 11, 1661--1681 (2011; Zbl 1246.22009)] studied the deformation when \(G\) stands for the Heisenberg group and showed that the Hausdorff property of the deformation space is equivalent to the fact that \(\varphi \in\mathcal R(G, \Gamma, H)\) is open in \(\mathrm{Hom}(\Gamma, G)\) (which means that the stability property holds). In this work, the authors study the stability property of discontinuous groups of connected threadlike Lie groups \(G\) of dimension \(n+1(n\in \mathbb N)\). Threadlike means here that the Lie algebra \(\mathfrak g_n\) of \(G\) admits a basis \(\mathcal B_n= \{ X, Y_1,\cdots, Y_n\}\) with Lie brackets: \[ [X, Y_i]= Y_{n+1}, \quad \forall i \in \{1, \cdots, n\}. \] They prove that if \(G\) is a threadlike group, then any non-abelian discrete subgroup of \(G\) is stable. When \(\Gamma\) is abelian, the property of stability fails to hold in general and depends upon the structure and the position of \(H\) and \(\Gamma\) inside \(G\). For the entire collection see [Zbl 1392.42001].