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**Classification of semisimple symmetric spaces with proper \(\mathrm{SL}(2,\mathbb R)\)-actions.**
*(English)*
Zbl 1312.53076

The motivation behind this paper is to address a problem introduced in [T. Kobayashi, “Discontinuous groups for non-Riemannian homogeneous spaces”, in: Mathematics unlimited – 2001 and beyond. Berlin: Springer. 723–747 (2001; Zbl 1023.53031)] namely “Fix a simply connected symmetric space \(\tilde{M}\) as a model space. What discrete groups can arise as the fundamental groups of complete affine manifolds \(M\) which are locally isomorphic to the space \(\tilde{M}\)?” which can rephrased as “Fix a symmetric pair \((G,H)\). What discrete groups does \(G/H\) admit as discontinuous groups?”

It is known that if there exists a Lie group homorphism \(\Phi: \mathrm{SL}(2,{\mathbb R}) \mapsto G\) such that \(\mathrm{SL}(2,{\mathbb R})\) acts on \(G/H\) properly via \(\Phi\) then the space \(G/H\) admits an infinite discontinuous group. The author sets out to prove the converse when \(G/H\) is a semisimple symmetric space.

Indeed, he proves two main results. In the first one, for a connected linear semisimple Lie group \(G\), he provides four criteria (expanded to nine later in the text) that are equivalent for a symmetric pair \((G,H)\) to possess a Lie group homomorphism \(\Phi: \mathrm{SL}(2,{\mathbb R}) \mapsto G\) such that \(\mathrm{SL}(2,{\mathbb R})\) acts on \(G/H\) properly via \(\Phi\). In the second main result, he provides a complete list of pairs \((\mathfrak g,\mathfrak h)\) such that these equivalent conditions hold.

The criterion found in [Y. Benoist, Ann. Math. (2) 144, No. 2, 315–347 (1996; Zbl 0868.22013)] and the one found in the paper by Kobayashi cited above play an important role in the proofs.

The mains results of this paper were announced in [T. Okuda, Proc. Japan Acad., Ser. A 87, No. 3, 35–39 (2011; Zbl 1221.22020)] with sketches of proofs.

It is known that if there exists a Lie group homorphism \(\Phi: \mathrm{SL}(2,{\mathbb R}) \mapsto G\) such that \(\mathrm{SL}(2,{\mathbb R})\) acts on \(G/H\) properly via \(\Phi\) then the space \(G/H\) admits an infinite discontinuous group. The author sets out to prove the converse when \(G/H\) is a semisimple symmetric space.

Indeed, he proves two main results. In the first one, for a connected linear semisimple Lie group \(G\), he provides four criteria (expanded to nine later in the text) that are equivalent for a symmetric pair \((G,H)\) to possess a Lie group homomorphism \(\Phi: \mathrm{SL}(2,{\mathbb R}) \mapsto G\) such that \(\mathrm{SL}(2,{\mathbb R})\) acts on \(G/H\) properly via \(\Phi\). In the second main result, he provides a complete list of pairs \((\mathfrak g,\mathfrak h)\) such that these equivalent conditions hold.

The criterion found in [Y. Benoist, Ann. Math. (2) 144, No. 2, 315–347 (1996; Zbl 0868.22013)] and the one found in the paper by Kobayashi cited above play an important role in the proofs.

The mains results of this paper were announced in [T. Okuda, Proc. Japan Acad., Ser. A 87, No. 3, 35–39 (2011; Zbl 1221.22020)] with sketches of proofs.

Reviewer: Patrice Sawyer (Sudbury)

### MSC:

53C35 | Differential geometry of symmetric spaces |

53C30 | Differential geometry of homogeneous manifolds |

57S30 | Discontinuous groups of transformations |