Proper action on a homogeneous space of reductive type

Kobayashi, Toshiyuki

doi:10.1007/bf01443517

Cited by 126 publications

(180 citation statements)

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“…In view of observation 2.2, Kobayashi [7] posed the following analogous problems in a continuous setting.…”

Section: Problems In a Continuous Analoguementioning

confidence: 99%

“…Loosely speaking, this phenomenon indicates that the fundamental group of a manifold might be not very complicated if is locally isomorphic to such a homogeneous manifold / . The first example for the Calabi-Markus phenomenon was found by Calabi and Markus in 1962 [3] for ( , ) = ( ( , 1), ( − 1,1)), and generalized by Wolf [22] and Kulkarni [15] for ( , ) = ( ( , ), ( ; − 1)) from 1960s to the early 1980s, and then completely settled by Kobayashi [7] in 1989 for reductive Lie groups ( , ) in terms of real rank conditions. The key lemma of Kobayashi's paper [7] is to establish the criterion of proper actions of continuous subgroups.…”

Section: Introductionmentioning

confidence: 97%

“…The first example for the Calabi-Markus phenomenon was found by Calabi and Markus in 1962 [3] for ( , ) = ( ( , 1), ( − 1,1)), and generalized by Wolf [22] and Kulkarni [15] for ( , ) = ( ( , ), ( ; − 1)) from 1960s to the early 1980s, and then completely settled by Kobayashi [7] in 1989 for reductive Lie groups ( , ) in terms of real rank conditions. The key lemma of Kobayashi's paper [7] is to establish the criterion of proper actions of continuous subgroups. Here the notion of proper action was proposed by Palais [19] in 1961to single out a nice category of actions of non-compact groups.…”

Section: Introductionmentioning

confidence: 97%

“…Since Kobayashi's paper [7] in 1989, there have been extensive studies of CliffordKlein forms of homogeneous spaces / for a reductive Lie group by various methods (see surveys [10,11] and references therein). On the other hand, much has not been studied in the case where is a nilpotent Lie group, which is supposed to be an opposite extremal case of reductive groups.…”

Section: Introductionmentioning

confidence: 99%

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Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

Salma

2012

J. Sci. Res.

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Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989. On the other hand, an analogous criterion to Kobayashi's equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups and are equivalent: (i) the action of on / is proper; (ii) ⋂ −1 is compact for any ∊ . The condition (i) is important in the study of discontinuous groups for the homogeneous space / , while the second condition (ii) can easily be checked. The implication (i) (ii) is obvious, and the opposite implication (ii) (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ⇔ (ii) for certain affine nilpotent Lie groups .

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“…In view of observation 2.2, Kobayashi [7] posed the following analogous problems in a continuous setting.…”

Section: Problems In a Continuous Analoguementioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

Salma

2012

J. Sci. Res.

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“…Le théorème 1.1 est également valable pour k = R d'après [15], [12] et [14]. Les arguments de [14] utilisent la dimension cohomologique de Γ, TOME 60 (2010), FASCICULE 5 (G × G)/∆ G et qui sont Zariski-denses dans G × G. On peut de plus choisir Γ de telle sorte qu'aucune de ses deux projections naturelles sur G ne soit bornée.…”

Section: éNoncé Des Résultats Principauxunclassified