scholarly journals Orbit equivalence and rigidity of ergodic actions of Lie groups

1981 ◽  
Vol 1 (2) ◽  
pp. 237-253 ◽  
Author(s):  
Robert J. Zimmer
Keyword(s):  
Invariant Measure ◽  
Euclidean Space ◽  
Lie Groups ◽  
Homogeneous Spaces ◽  
The Other ◽  
Rigidity Theorem ◽  
Simple Groups ◽  
Orbit Equivalence ◽  
Finite Invariant Measure ◽  
Lattice Subgroups

AbstractThe rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

Ergodic elements for actions of Lie groups

1996 ◽  
Vol 16 (4) ◽  
pp. 703-717
Author(s):  
K. Robert Gutschera
Keyword(s):  
Invariant Measure ◽  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Group Structure ◽  
Structure Theory ◽  
Ergodic Action ◽  
Finite Invariant Measure ◽  
Connected Lie Group

AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

On an extension of Wallace's pedal property of the circumcircle

1924 ◽  
Vol 22 (1) ◽  
pp. 34-41 ◽  
Author(s):  
H. W. Richmond
Keyword(s):  
Euclidean Space ◽  
The Other ◽  
Analogous Property ◽  
Recent Part

1. Mr J. P. Gabbatt has discussed in the most recent Part of the Proceedingsof this Society the Pedal locus of a simplex in hyperspace. It is, however, possible to regard the pedal property of the circumcircle somewhat differently and so to seek other extensions. Given a circle, any three points on it are vertices of an inscribed triangle, and the feet of the perpendiculars on the sides from any fourth point of the circle are collinear. Is there any curve in space on which an analogous property holds for any five points, viz. that the feet of the perpendiculars from any one upon the faces of the tetrahedron formed by the other four are coplanar?It will be shown that curves of order n exist in Euclidean space of n dimensions on which any n + 2 points have such a property; but that the curves cannot be real if n is odd.

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.

On Finite Invariant Measure for Semigroups of Operators

1971 ◽  
Vol 14 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Usha Sachdevao
Keyword(s):  
Invariant Measure ◽  
Semigroups Of Operators ◽  
Amenable Semigroup ◽  
Linear Contraction ◽  
Finite Invariant Measure

Let Σ be a left amenable semigroup, and let {Tσ: σ ∊ Σ} be a representation of Σ as a semigroup of positive linear contraction operators on L1(X, 𝓐, p). This paper is devoted to the study of existence of a finite equivalent invariant measure for such semigroups of operators.

scholarly journals On completeness of holomorphic principal bundles

1975 ◽  
Vol 56 ◽  
pp. 121-138 ◽  
Author(s):  
Shigeru Takeuchi
Keyword(s):  
Algebraic Group ◽  
Lie Groups ◽  
Abelian Variety ◽  
Lie Group ◽  
Basic Structure ◽  
Factor Group ◽  
The Other ◽  
Algebraic Subgroup ◽  
Points Of View ◽  
Lie Subgroup

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

Representations of Lie Groups by Contact Transformations, II: Non-Compact Simple Groups

1993 ◽  
Vol 45 (4) ◽  
pp. 778-802
Author(s):  
Carl Herz
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Compact Manifold ◽  
Transitive Group ◽  
Simple Groups ◽  
Representations Of Lie Groups ◽  
Contact Transformations

AbstractIf a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described

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