Proper SL(2, ℝ)-actions on homogeneous spaces

Abstract

We study the existence problem of proper actions of SL(2, R) on homogeneous spaces G/H of reductive type. Based on Kobayashi's properness criterion [Math. Ann. (1989)], we show that G/H admits a proper SL(2, R)-action via G if a maximally split abelian subspace of Lie H is included in the wall defined by a restricted root of Lie G. We also give a number of examples of such G/H.

Anosov representations and proper actions

“…, a d ∈ R} be the set of diagonal matrices contained in g, where G = SO (d, d) is defined by the quadratic form x 1 x d+1 + · · · + x d x 2d . Let Table 1 were obtained by checking the condition of Lemma 6.1 in a list of examples of nonsymmetric G/H admitting properly discontinuous actions by nonvirtually abelian groups given in [BJT14].…”

“…In examples (i) to (iv) when k = d/2, in example (v) when ℓ = d/4, and in example (vii), the group G 1 acts cocompactly on G/H, hence compact Clifford-Klein forms of G/H can be obtained by taking Γ to be a uniform lattice in G 1 . We refer to [Oku13,BJT14] for many more examples to which Corollary 1.12 applies, where G 1 is locally isomorphic to SL 2 (R). Table 3.…”

Nonexistence of standard compact Clifford–Klein forms of homogeneous spaces of exceptional Lie groups

“…We have also created a special plugin CKForms [3] which uses the following plugins: SLA [13], CoReLG [9]. 4 Analyzing Table 1 and the second step of proof of Theorem 1…”

“…There are many partial results which yield a strong evidence for the conjecture. For instance there are topological obstructions [24], [28], [33], results of T. Kobayashi [21], [22], [23], Margulis [26], Zimmer [35], Hee Oh and Witte [31], Okuda [32] and the authors of this paper [2], [4], [6], [7]. However, Kobayashi's question (which is challenging and mathematically interesting) is still to be answered.…”

Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a calculational approach

“…Using the a-hyperbolic ranks one can formulate conditions (similar to the Calabi-Markus phenomenon) under which the space G/H admits, or does not admit proper actions of non-virtually abelian discrete subgroups ([3], Theorem 2). In [7] other examples of homogeneous spaces G/H admitting proper actions of non-virtually abelian discrete subgroups were given.…”