Cartan projections of some non-reductive subgroups and proper actions on homogeneous spaces (Q6594720)

Let \(G\) be a Lie group and \(H\) be a closed Lie subgroup of \(G\). If some discrete subgroup \(\Gamma \subset G\) acts properly and freely on \(G/H\), the quotient space is called a Clifford-Klein form for \(G/H\). Given a homogeneous space \(G/H\), one can consider the problem to determine whether it admits a compact Clifford-Klein form or not. This problem has been studied in many papers, especially when \(G/H\) is a homogeneous space of reductive type and the subgroup \(H\) is non-compact.\N\NThe main result of this paper, which is based on some general result by \textit{T. Kobayashi} [Duke Math. J. 67, No. 3, 653--664 (1992; Zbl 0799.53056)], is the following. Let \(K = \mathbb R, \mathbb C\) or \(\mathbb H\). If \(m \ge 2\) and \(n \ge 5m/4 + \epsilon (m, K)\) (the function \(\epsilon (m,K)\) is fully described in the paper), then \(G/H = \mathrm{SL}(n, K)/ \mathrm{SL}(m, K)\) does not admit a compact Clifford-Klein form. Also a new proof of the following result by the author [Sel. Math., New Ser. 23, No. 3, 1931--1953 (2017; Zbl 1372.57043)] is given here: Let \(G/H\) be a reductive symmetric space and \((g, h)\) be the corresponding reductive symmetric pair. Assume that \(H\) is non-compact and the associated symmetric subalgebra is a Levi subalgebra of \(g\). Then \(G/H\) does not admit a compact Clifford-Klein form. As an example it follows that if \(p,q \ge 1\), then \(\mathrm{SL}(p + q, {\mathbb R})/ \mathrm{SO}_0(p, q)\) does not admit a compact Clifford-Klein form.