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**Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds.**
*(English)*
Zbl 1026.57029

This article is based on lectures delivered by the author at the 2000 Twente Conference on Lie groups. Their goal was “to give an accessible exposition by clarifying the current status of some of the central problems of this topic”. The following two problems for homogeneous spaces \(G/H\), where \(G\) and \(H\) are reductive real linear noncompact groups, are considered: A) find a criterion for a discrete subgroup \(\Gamma\subset G\) to act properly discontinuously on \(G/H\); B) determine pairs \(G,H\) such that \(G/H\) admits a subgroup \(\Gamma\subset G\) whose action on \(G/H\) is cocompact, properly discontinuous, and free (in this case, \(\Gamma\backslash G/H\) is said to be a Clifford-Klein form for \(G/H\)).

Solutions to A) were independently given by the author and Benoist. Any real reductive linear group admits the Cartan decomposition \(G=KAK\); the Cartan projection \(\nu\) is defined by \(\nu(g)=\log a\pmod W\) for \(g=k_1ak_2\), where \(W\) is the Weyl group of \(G\). The mapping \(\nu\) keeps two special relations between subsets \(L,H\) of groups: a) \(L\mathbin\cap SHS\) is relatively compact for any compact set \(S\); b) there exists a compact \(S\) such that \(L\subseteq SHS\) and \(H\subseteq SLS\). If \(L\) and \(H\) are closed subgroups of \(G\) then \(L\) acts properly on \(G/H\) if and only if they satisfy a). On the other hand, it is easy to work with these relations in the abelian group \(A\). Also, there is a kind of duality between a) and b). In particular, these facts yield the Calabi-Marcus phenomenon: the absence of infinite discontinuous groups on \(G/H\) is equivalent to the condition that real ranks of \(G\) and \(H\) are equal.

For B), there is no complete solution. If \(H\) is compact then a compact Clifford-Klein form for \(G/H\) exists if and only if \(G\) contains a cocompact discrete subgroup \(\Gamma\); according to Borel’s theorem, this is true for all real reductive linear groups (moreover, \(\Gamma\) can be assumed to be arithmetic). For noncompact \(H\), a method for constructing these forms is explained. It includes three steps: 1) take a connected subgroup \(L\) acting properly and cocompactly on \(G/H\); 2) take a cocompact and torsion free discrete subgroup \(\Gamma\) in \(L\); 3) deform \(\Gamma\) in \(G\). For a reductive linear group \(G\), set \(d(G)=\dim G -\dim K\), where \(K\) is the maximal compact subgroup of \(G\). If there exists a reductive subgroup \(L\) of the reductive linear group \(G\) such that \(L,H\) satisfy a) and \(d(L)+d(H)=d(G)\), then \(G/H\) admits a compact Clifford-Klein form. If \(H,L\) are stable with respect to some Cartan involution and \(H\mathbin\cap L\) is compact, then the equality \(\dim H+\dim L=\dim G+\dim(H\mathbin\cap L)\) is sufficient. A list of spaces \(G/H\) that admit compact Clifford-Klein forms obtained this way is given. Necessary conditions for the existence of these forms are also discussed. The simple one is the condition that the real rank of \(H\) is strictly less than the real rank of \(G\) (this follows from the Calabi-Marcus phenomenon). If \(\nu(L)\subseteq\nu(H)\) (up to relation b) above) and \(d(L)>d(H)\) for some reductive subgroup \(L\), then \(G/H\) does not admit a compact Clifford-Klein form. Some results of Benoist, Margulis, Zimmer and other authors are described. No complete proof is given. Some open problems and conjectures are formulated.

Solutions to A) were independently given by the author and Benoist. Any real reductive linear group admits the Cartan decomposition \(G=KAK\); the Cartan projection \(\nu\) is defined by \(\nu(g)=\log a\pmod W\) for \(g=k_1ak_2\), where \(W\) is the Weyl group of \(G\). The mapping \(\nu\) keeps two special relations between subsets \(L,H\) of groups: a) \(L\mathbin\cap SHS\) is relatively compact for any compact set \(S\); b) there exists a compact \(S\) such that \(L\subseteq SHS\) and \(H\subseteq SLS\). If \(L\) and \(H\) are closed subgroups of \(G\) then \(L\) acts properly on \(G/H\) if and only if they satisfy a). On the other hand, it is easy to work with these relations in the abelian group \(A\). Also, there is a kind of duality between a) and b). In particular, these facts yield the Calabi-Marcus phenomenon: the absence of infinite discontinuous groups on \(G/H\) is equivalent to the condition that real ranks of \(G\) and \(H\) are equal.

For B), there is no complete solution. If \(H\) is compact then a compact Clifford-Klein form for \(G/H\) exists if and only if \(G\) contains a cocompact discrete subgroup \(\Gamma\); according to Borel’s theorem, this is true for all real reductive linear groups (moreover, \(\Gamma\) can be assumed to be arithmetic). For noncompact \(H\), a method for constructing these forms is explained. It includes three steps: 1) take a connected subgroup \(L\) acting properly and cocompactly on \(G/H\); 2) take a cocompact and torsion free discrete subgroup \(\Gamma\) in \(L\); 3) deform \(\Gamma\) in \(G\). For a reductive linear group \(G\), set \(d(G)=\dim G -\dim K\), where \(K\) is the maximal compact subgroup of \(G\). If there exists a reductive subgroup \(L\) of the reductive linear group \(G\) such that \(L,H\) satisfy a) and \(d(L)+d(H)=d(G)\), then \(G/H\) admits a compact Clifford-Klein form. If \(H,L\) are stable with respect to some Cartan involution and \(H\mathbin\cap L\) is compact, then the equality \(\dim H+\dim L=\dim G+\dim(H\mathbin\cap L)\) is sufficient. A list of spaces \(G/H\) that admit compact Clifford-Klein forms obtained this way is given. Necessary conditions for the existence of these forms are also discussed. The simple one is the condition that the real rank of \(H\) is strictly less than the real rank of \(G\) (this follows from the Calabi-Marcus phenomenon). If \(\nu(L)\subseteq\nu(H)\) (up to relation b) above) and \(d(L)>d(H)\) for some reductive subgroup \(L\), then \(G/H\) does not admit a compact Clifford-Klein form. Some results of Benoist, Margulis, Zimmer and other authors are described. No complete proof is given. Some open problems and conjectures are formulated.

Reviewer: V.Gichev (Omsk)

### MSC:

57S30 | Discontinuous groups of transformations |

22F30 | Homogeneous spaces |

22E40 | Discrete subgroups of Lie groups |

32M10 | Homogeneous complex manifolds |

53C30 | Differential geometry of homogeneous manifolds |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |