Narrow Results

We consider the affine action of a nilpotent Lie group on ℝⁿ. Lipsman (1995) conjectured that such an action is proper in the sense of Palais if and only if the action is (CI) in the sense of Kobayashi. The present paper gives a counterexample to Lipsman's conjecture for n ≥ 5.

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a 'small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of discontinuous groups, we introduce the concepts from a group theoretic perspective, and focus on 'stability' and 'local rigidity' of discontinuous groups. As a test case, we give an explicit description of the deformation space of ℤ^k acting properly discontinuously on ℝ^k+1 by affine nilpotent transformations. Our method uses an idea of 'continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

The Cartan motion group associated to a Riemannian symmetric space X is a semidirect product group acting isometrically on its tangent space. For two subsets in a locally compact group G, Kobayashi introduced the concept of "properness" as a generalization of properly discontinuous actions of discrete subgroups on homogeneous spaces of G. In this paper, we give a criterion of properness for homogeneous spaces of Cartan motion groups. Our criterion has a similar feature to the case where G is a reductive Lie group.

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

For a Lie group G and a subset H of G, Kobayashi [4] introduced the concept of the discontinuous dual of H as the set consisting of the subsets in G which are proper with H. Here, properness is a generalization of properly discontinuous actions. Kobayashi posed in the millenium volume [5] whether the discontinuous dual of H determines H uniquely up to a certain equivalence relation. In this paper, we solve this problem affirmatively.

Let H be a closed connected subgroup of a connected, simply connected exponential solvable Lie group G. We consider the deformation space of a discontinuous subgroup Γ of G for the homogeneous space G/H. When H contains [G, G], we exhibit a description of the space which appears to involve GL_k(ℝ) as a direct product factor, where k designates the rank of Γ. The moduli space is also described. Consequently, we prove in such a setup that the local rigidity property fails to hold globally on and that every element of the parameters space is topologically stable.

Let G = H_{2n + 1} be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤ^k and ℝ^k+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space $G/H$ admits a discontinuous group which is not virtually abelian if and only if $G/H$ admits a proper $SL(2,ℝ)$-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces $G/H$ of reductive type. We give a counterexample in the case $G=SL(5,ℝ)$.

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space $G/H$, determining explicitly which homogeneous spaces $G/H$ allow nontrivial continuous deformations of co-compact discontinuous groups. When $G$ is assumed to be exponential solvable and $H\subset G$ is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if $G$ is isomorphic to the group Aff($ℝ$) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not.17 (2012) 4062–4084.]). The present paper deals with the more general context, when $G$ is a connected solvable Lie group and $H$ a maximal nonnormal subgroup of $G$. We prove that any discontinuous group $\mathrm{\Gamma }$ for a homogeneous space $G/H$ is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the $G$-action on $G/H$ is not always effective, and thus the space of group theoretic deformations (formal deformations) $\mathcal{𝒯}(\mathrm{\Gamma },G;G/H)$ could be larger than geometric deformation spaces. We determine $\mathcal{𝒯}(\mathrm{\Gamma },G;G/H)$ and also its quotient modulo uneffective parts when the rank $\mathrm{\Gamma }=1$. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Let $G$ be a Lie group, $H$ a closed subgroup of $G$ and $\mathrm{\Gamma }$ a discontinuous group for the homogeneous space $\mathcal{𝒳}=G/H$. Given a deformation parameter $\phi \in \mathrm{\text{Hom}}(\mathrm{\Gamma },G)$, the deformed subgroup $\phi (\mathrm{\Gamma })$ may fail to act properly discontinuously on $\mathcal{𝒳}$. To understand this phenomenon in the case when $G$ stands for an Euclidean motion group $O_n(ℝ)⋉{ℝ}^n$, we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of ${\mathbb{Z}}^k$ on ${ℝ}^{k+1}$, Int. J. Math.17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when $\mathrm{\Gamma }$ turns out to be a crystallographic subgroup of $G$.

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group $\mathrm{\Gamma }\subset G_n$ acting on a homogeneous space $G_n/H$, where $H$ stands for a closed subgroup of the Heisenberg motion group $G_n:={𝕌}_n⋉{ℍ}_n$. That is, the parameter space admits a locally rigid (equivalently a strongly locally rigid) point if and only if $\mathrm{\Gamma }$ is finite. Moreover, Calabi–Markus’s phenomenon and the question of existence of compact Clifford–Klein forms are also studied.

In this paper, we discuss local rigidity of Clifford–Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. We split the property of local rigidity into two conditions: vertical rigidity and horizontal rigidity. By this separation, we discuss local rigidity, in particular, Baklouti’s conjecture.

In $1964$, Auslander conjectured that every crystallographic subgroup $\mathrm{\Gamma }$ of the affine group $G{L}_n(ℝ)⋉{ℝ}^n$ acting properly discontinuously on ${ℝ}^n$ is virtually solvable, i.e. contains a solvable subgroup of finite index. One of the major difficulties to prove such a conjecture is to generate a criterion of the proper action of a given discrete group acting properly discontinuously on ${ℝ}^n$. In this paper, we focus on this question in the setting where $\mathrm{\Gamma }$ is abelian. We show that $\mathrm{\Gamma }\simeq {\mathbb{Z}}^n$ acts properly on ${ℝ}^n$ if and only if any $\mathrm{\Gamma }$-orbit is discrete (or equivalelntly closed). We further generate an equivalent criterion in the setting where $\mathrm{\Gamma }$ stands for a crystallographic discontinuous affine group for ${ℝ}^n$ based upon a generating family.

Let K be a compact subgroup of $GL(n,ℝ)$ and $G:=K⋉{ℝ}^n$ the semi-direct product group. Let H be a closed subgroup of G and $\mathrm{\Gamma }$ a discontinuous group for the homogeneous space $\mathcal{𝒳}=G/H$. We establish a geometrical criterion of the proper action of $\mathrm{\Gamma }$ on $\mathcal{𝒳}$, which requires an accurate description of the structure of closed connected subgroups of Euclidean motion groups. As a consequence, we establish a criterion for the existence of a compact Clifford–Klein form $\mathrm{\Gamma }\backslash \mathcal{𝒳}$ and study the Calabi–Markus phenomenon.

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes Elfving’s dissertation theorem for locally linear $G$-manifolds (1996). Also, we improve the Bredon–Floyd theorem from compact Lie groups $G$ to arbitrary Lie groups $G$.

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.

We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.