Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford–Klein forms problem

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

“…Therefore for every potential triple (e 7 (7) , e 6(2) ⊕so(2), l) (that is a triple that fulfills the requirements 2. and 3. of Proposition 3) there exists a real hyperbolic orbit in g that meets l and e 6(2) (we use a fact that a hyperbolic orbit in g = k + p always meets p, and thus every hyperbolic orbit meeting e 6(2) ⊕ so(2) meets e 6(2) ). But by Corollary 2 this orbit meets every copy of e 6 (2) . Thus the cases 7, 8 can not yield standard compact Clifford-Klein forms.…”

“…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.…”

Non-virtually abelian discontinuous group actions vs. proper SL(2,R)-actions on homogeneous spaces

“…This is a long standing open problem [20,21,22]. There are many partial results, for example, [4,5,6,9,25], however, the general case is not settled.…”

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

“…For example, in [4] we proved that there are no compact Clifford-Klein forms of exceptional Lie groups. In [6] we proposed an algorithm for checking an obstruction to the existence of compact Clifford-Klein forms. Looking at the problem from this angle we keep in mind that there are algorithms for various calculations in semisimple Lie algebras designed by de Graaf et.…”