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.

ON THE DEFORMATION SPACE OF CLIFFORD–KLEIN FORMS OF SOME EXPONENTIAL HOMOGENEOUS SPACES

2009 ◽  
Vol 20 (07) ◽  
pp. 817-839 ◽  
Author(s):  
ALI BAKLOUTI ◽  
IMED KÉDIM
Keyword(s):  
Moduli Space ◽  
Homogeneous Spaces ◽  
Deformation Space ◽  
Local Rigidity ◽  
Connected Subgroup ◽  
Rigidity Property ◽  
Product Factor ◽  
Closed Connected Subgroup ◽  
Simply Connected ◽  
Solvable Lie Group

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

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.

On the conjugacy and isomorphism problems for stabilizers of Lie group actions

1999 ◽  
Vol 19 (2) ◽  
pp. 391-411 ◽  
Author(s):  
VALENTIN YA. GOLODETS ◽  
SERGEY D. SINEL'SHCHIKOV
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Group Actions ◽  
Transformation Groups ◽  
Type I ◽  
Lie Group Actions ◽  
The Stability ◽  
Regular Transformation

The spaces of subgroups and Lie subalgebras with the group actions by conjugations are considered for real Lie groups. Our approach allows one to apply the properties of algebraically regular transformation groups to finding the conditions when those actions turn out to be type I. It follows, in particular, that in this case the stability groups for all the ergodic actions of such groups are conjugate (for example when the stability groups are compact). The isomorphism of the stability groups for ergodic actions is also established under some assumptions. A number of examples of non-conjugate and non-isomorphic stability groups are presented.

scholarly journals Analytic representation theory of Lie groups: general theory and analytic globalizations of Harish-Chandra modules

2011 ◽  
Vol 147 (5) ◽  
pp. 1581-1607 ◽  
Author(s):  
Heiko Gimperlein ◽  
Bernhard Krötz ◽  
Henrik Schlichtkrull
Keyword(s):  
General Theory ◽  
Representation Theory ◽  
Lie Groups ◽  
Lie Group ◽  
Analytic Functions ◽  
General Framework ◽  
Analytic Representation ◽  
Reductive Groups ◽  
Topological Properties ◽  
Rapid Decay

AbstractIn this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra 𝒜(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and 𝒜(G) and which embeds as the space of analytic vectors in all Banach globalizations of V.

Homogeneous Complex Manifolds with more than One End

1989 ◽  
Vol 41 (1) ◽  
pp. 163-177 ◽  
Author(s):  
B. Gilligan ◽  
K. Oeljeklaus ◽  
W. Richthofer
Keyword(s):  
Lie Group ◽  
Homogeneous Spaces ◽  
Maximal Compact Subgroup ◽  
Compact Subgroup ◽  
Complex Manifolds ◽  
Connected Subgroup ◽  
Homogeneous Complex ◽  
Homogeneous Complex Manifold ◽  
Homogeneous Cone ◽  
Closed Connected Subgroup

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

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.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350055
Author(s):  
SONIA L'INNOCENTE ◽  
FRANÇOISE POINT ◽  
CARLO TOFFALORI
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Model Theory ◽  
Effective Model ◽  
Logarithmic Function ◽  
Compact Lie Groups ◽  
Model Completeness ◽  
Schanuel's Conjecture ◽  
Logarithm Function ◽  
Natural Expansion

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

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