## Proper actions on reductive homogeneous spaces. (Actions propres sur les espaces homogènes réductifs.)(French)Zbl 0868.22013

Let $$G$$ be a linear semisimple real Lie group and let $$H$$ be a reductive subgroup. This paper gives necessary and sufficient conditions for the existence of a nonabelian free subgroup $$\Gamma$$ of $$G$$ that acts properly on $$G/H$$. In particular, the author shows that such a group $$\Gamma$$ does not exist for $$SL(2n+1,R)/SL(2n,R)$$. When $$\Gamma$$ does not exist, it follows that $$G/H$$ does not have a compact quotient (i.e., there is no discrete subgroup $$\Gamma$$ acting properly on $$G/H$$ such that $$\Gamma \backslash G/H$$ is compact). It follows that $$SL(2n+1,R)/SL(2n,R)$$ does not have a compact quotient, which has also been shown by Margulis. Moreover, the case of complex symmetric spaces is studied extensively. We state the main theorem more precisely. Let $$A_H$$ be a maximal split abelian subgroup of $$H$$ and let $$A$$ be a maximal split abelian subgroup of $$G$$ containing $$A_H.$$ Let $$A^+$$ be a Weyl chamber and let $$B=\{ x\in A^+ |w_0(x)=x^{-1} \}$$, where $$w_0$$ is the long element of the Weyl group of $$G.$$ The main theorem asserts that there is a discrete subgroup $$\Gamma$$ which does not contain an abelian subgroup of finite index acting properly on $$G/H$$ if and only if $$wA_H$$ does not contain $$B$$ for any $$w$$ in the Weyl group. Moreover, one may assume $$\Gamma$$ acts freely on $$G/H$$ and is Zariski dense in $$G.$$ The condition is easy to check in examples. It is known that $$G/H$$ contains an infinite discrete subgroup acting properly on $$G/H$$ if and only if $$A_H\not= A,$$ so we may as well assume $$A_H\not= A.$$ Note that $$A^+=B$$ whenever $$w_0(x)=x^{-1},$$ which happens except for $$A_n, n\geq 2, D_{2n+1},$$ and $$E_6.$$ When $$A^+=B$$ and $$A_H\not= A,$$ then of course, $$wA_H$$ does not contain $$B$$ for any $$w$$ by reasons of dimension. The main technique of the proof is to study the Cartan projection of a discrete group onto the dominant Weyl chamber. To show that no $$\Gamma$$ action exists in the main theorem, the author produces infinitely many elements in a subset of the positive Weyl chamber. The author deduces the results about compact quotients by showing that a quotient by a nilpotent discrete group cannot be compact via a reduction to the rank one case. The main results hold for groups over any local field of characteristic zero.
Reviewer: S.Evens (Tucson)

### MSC:

 22E15 General properties and structure of real Lie groups 43A85 Harmonic analysis on homogeneous spaces 53C30 Differential geometry of homogeneous manifolds 22E40 Discrete subgroups of Lie groups
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