Criterion of Proper Actions for 3-Step Nilpotent Lie Groups

Yoshino, Taro

doi:10.1142/s0129167x07004333

Cited by 11 publications

(6 citation statements)

References 9 publications

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“…The present setup appears therefore as a new instance to ascertain these facts as shown in the papers [1,3,19]. On the other hand and according to the works [4,16,24,26,28,29], the proper action of a connected Lie subgroup L on nilpotent homogeneous spaces is equivalent to its free action whenever the nilpotent Lie group in question is of step strictly less than four. This fact greatly contributes to simplifying the explicit determination of the parameters and the deformation spaces as we get accessible means to meddle with the proper action.…”

supporting

confidence: 52%

“…In [1], the following characterization of the parameter and the deformation spaces is derived as follows. Let L be the syndetic hull of which is the smallest (and hence the unique) connected Lie subgroup of G which contains cocompactly (see [8] and [29]). Recall that the Lie subalgebra l of L is the real span of the abelian lattice log , which is generated by {log γ 1 , .…”

Section: Characterization Of the Parameter And Deformation Spacesmentioning

confidence: 99%

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Deforming discontinuous subgroups for threadlike homogeneous spaces

Baklouti

Khlif

2009

Geom Dedicata

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Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous subgroup for the homogeneous space M = G/H and any deformation of , the deformed discrete subgroup may utterly destroy its proper discontinuous action on M as H is not compact (except the case when it is trivial). To understand this specific issue, we provide an explicit description of the parameter and the deformation spaces of any abelian discrete acting properly discontinuously and fixed point freely on G/H for an arbitrary H of a threadlike nilpotent Lie group G. The topological features of deformations, such as the local rigidity and the stability are also discussed. Whenever the Clifford-Klein form \G/H in question is assumed to be compact, these spaces are cutely determined and unlike the case of Heisenberg groups, the deformation space fails in general to be a Hausdorff space. We show further that this space admits a smooth manifold as its open dense subset.

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supporting

confidence: 52%

Section: Characterization Of the Parameter And Deformation Spacesmentioning

confidence: 99%

Deforming discontinuous subgroups for threadlike homogeneous spaces

Baklouti

Khlif

2009

Geom Dedicata

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“…About this conjecture, the following results have been obtained so far. The properness and the property (CI) is equivalent for less than or equal to 3-step nilpotent Lie groups [28], [32] and not equivalent for 4-step nilpotent Lie groups [32]. In this subsection, we generalize Nasrin's result.…”

Section: Properness and (Ci) Propertymentioning

confidence: 70%

“…There have been attempts to extend Kobayashi's theory on discontinuous groups for reductive cases [13][14][15][16][17][18][19] to non-reductive cases such as Baklouti-Kédim [1], Kath-Olbrich [13], Kobayashi-Nasrin [21], Lipsman [27], Nasrin [28], Yoshino [32] and so on. In this section, we examine a 'solvable analogue' of Kobayashi's conjecture (Conjecture 1.4) and show an evidence that the assumption 'reductive type' in Kobayashi's conjecture is crucial.…”

Section: On Kobayashi's Conjecturementioning

confidence: 99%

Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature $(2, 2)$

Maeta¹

2020

Preprint

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We give a classification of irreducible four-dimensional symmetric spaces G/H which admit compact Clifford-Klein forms, where G is the transvection group of G/H. For this, we develop a method that applies to particular 1-connected solvable symmetric spaces.We also examine a 'solvable analogue' of Kobayashi's conjecture for reductive groups and find an evidence that the reductive assumption in Kobayashi's conjecture is crucial.

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“…3 Other recent activity in this program concerns the situation, where G is nilpotent and Γ is acting properly but not necessarily cocompact on G/H. (See, for example [6,7,46,80]. )…”

Section: Introductionmentioning

confidence: 99%

Deformation Spaces for Affine Crystallographic Groups

Baues

2008

Preprint

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We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study the properties of the action of the homotopy mapping class groups on deformation spaces. In our context these groups are arithmetic groups, and we construct examples of flat affine manifolds where every finite group of mapping classes admits a fixed point on the deformation space. We also show that the existence of fixed points on the deformation space is equivalent to the realisation of finite groups of homotopy equivalences by finite groups of affine diffeomorphisms. Extending ideas of Auslander we relate the deformation spaces of affine space forms with solvable fundamental group to deformation spaces of manifolds with nilpotent fundamental group. We give applications concerning the classification problem for affine space forms.

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Criterion of Proper Actions for 3-Step Nilpotent Lie Groups

Cited by 11 publications

References 9 publications

Deforming discontinuous subgroups for threadlike homogeneous spaces

Deforming discontinuous subgroups for threadlike homogeneous spaces

Four-dimensional compact Clifford-Klein forms of pseudo-Riemannian symmetric spaces with signature $(2, 2)$

Deformation Spaces for Affine Crystallographic Groups

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