scholarly journals DEFORMATION OF PROPERLY DISCONTINUOUS ACTIONS OF ℤk ON ℝk+1

2006 ◽  
Vol 17 (10) ◽  
pp. 1175-1193 ◽  
Author(s):  
TOSHIYUKI KOBAYASHI ◽  
SALMA NASRIN
Keyword(s):  
Small Deformation ◽  
Discrete Subgroup ◽  
Test Case ◽  
Deformation Space ◽  
Affine Transformations ◽  
Discontinuous Group ◽  
Proper Actions ◽  
Local Rigidity ◽  
Continuous Analogue ◽  
Discontinuous Groups

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 ℤk acting properly discontinuously on ℝk+1 by affine nilpotent transformations. Our method uses an idea of 'continuous analogue' and relies on the criterion of proper actions on nilmanifolds.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

2016 ◽  
Vol 27 (10) ◽  
pp. 1650085
Author(s):  
A. Baklouti ◽  
N. Elaloui ◽  
I. Kedim
Keyword(s):  
Homogeneous Space ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Rigidity Theorem ◽  
General Context ◽  
Affine Transformations ◽  
Discontinuous Group ◽  
Solvable Case ◽  
Local Rigidity ◽  
Discontinuous Groups

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) 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 [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] 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 [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. 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.

Deforming discontinuous groups for Heisenberg motion groups

2019 ◽  
Vol 30 (09) ◽  
pp. 1950045
Author(s):  
Ali Baklouti ◽  
Souhail Bejar ◽  
Khaireddine Dhahri
Keyword(s):  
Homogeneous Space ◽  
Parameter Space ◽  
Closed Subgroup ◽  
Motion Group ◽  
Discontinuous Group ◽  
Natural Action ◽  
Local Rigidity ◽  
Deformation Parameters ◽  
Motion Groups ◽  
Discontinuous Groups

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group [Formula: see text] acting on a homogeneous space [Formula: see text], where [Formula: see text] stands for a closed subgroup of the Heisenberg motion group [Formula: see text]. That is, the parameter space admits a locally rigid (equivalently a strongly locally rigid) point if and only if [Formula: see text] is finite. Moreover, Calabi–Markus’s phenomenon and the question of existence of compact Clifford–Klein forms are also studied.

Homogeneous space with non-virtually abelian discontinuous groups but without any proper SL(2, ℝ)-action

2016 ◽  
Vol 27 (03) ◽  
pp. 1650018
Author(s):  
Takayuki Okuda
Keyword(s):  
Symmetric Space ◽  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Powerful Method ◽  
Discrete Subgroups ◽  
Discontinuous Group ◽  
Semisimple Symmetric Space ◽  
Continuous Analogue ◽  
Discontinuous Groups

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 [Formula: see text] admits a discontinuous group which is not virtually abelian if and only if [Formula: see text] admits a proper [Formula: see text]-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 [Formula: see text] of reductive type. We give a counterexample in the case [Formula: see text].

Stability of discontinuous groups for reduced threadlike Lie groups

2015 ◽  
Vol 26 (08) ◽  
pp. 1550057 ◽  
Author(s):  
Fatma Khlif
Keyword(s):  
Lie Groups ◽  
Parameter Space ◽  
Lie Group ◽  
Deformation Space ◽  
Topological Properties ◽  
Local Rigidity ◽  
Connected Subgroup ◽  
The Stability ◽  
Closed Connected Subgroup ◽  
Discontinuous Groups

Let G be a reduced threadlike Lie group, H an arbitrary closed connected subgroup of G and Γ ⊂ G an abelian discontinuous subgroup for G/H. We study in this work some topological properties of the parameter space [Formula: see text] and the deformation space [Formula: see text], namely the stability and the rigidity. Instead of treating stability, we consider a weaker form by using an explicit covering of Hom (Γ, G) which we call layering and we show that the local rigidity holds if and only if Γ is finite.

scholarly journals Discontinuous groups of affine transformations of $C^{3}$

1976 ◽  
Vol 28 (1) ◽  
pp. 89-94 ◽  
Author(s):  
Kôji Uchida ◽  
Hisao Yoshihara
Keyword(s):  
Affine Transformations ◽  
Discontinuous Groups

Variants of stability of discontinuous groups for Euclidean motion groups

2017 ◽  
Vol 28 (06) ◽  
pp. 1750046 ◽  
Author(s):  
Ali Baklouti ◽  
Souhail Bejar
Keyword(s):  
Lie Group ◽  
Closed Subgroup ◽  
Deformation Parameter ◽  
Motion Group ◽  
Discontinuous Group ◽  
Euclidean Motion Group ◽  
Euclidean Motion ◽  
Motion Groups ◽  
Discontinuous Groups ◽  
Properly Discontinuous Action

Let [Formula: see text] be a Lie group, [Formula: see text] a closed subgroup of [Formula: see text] and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. Given a deformation parameter [Formula: see text], the deformed subgroup [Formula: see text] may fail to act properly discontinuously on [Formula: see text]. To understand this phenomenon in the case when [Formula: see text] stands for an Euclidean motion group [Formula: see text], we compare the notion of stability for discontinuous groups (cf. [T. Kobayashi and S. Nasrin, Deformation of properly discontinuous action of [Formula: see text] on [Formula: see text], Int. J. Math. 17 (2006) 1175–1193]) with its variants. We prove that the defined stability variants hold when [Formula: see text] turns out to be a crystallographic subgroup of [Formula: see text].

scholarly journals On Algebraic Groups and Discontinuous Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 279-322 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Discrete Subgroup ◽  
Simple Algebraic Group ◽  
Discontinuous Groups

Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

Sign in / Sign up

Export Citation Format

Share Document