Journal ArticleDOI
On Ergodic quasi-invariant measures of group automorphism
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In this paper, the authors studied the dynamics of projective transformations and applied it to prove that the isotropy subgroups of probability measures on algebraic homogeneous spaces are algebraic and study the class of ergodic quasi-invariant measures of automorphisms of non-compact Lie groups.Abstract:
We study the dynamics of projective transformations and apply it to (i) prove that the isotropy subgroups of probability measures on algebraic homogeneous spaces are algebraic and to (ii) study the class of ergodic quasi-invariant measures of automorphisms of non-compact Lie groups. It is shown that their support is always a proper subset and that under certain conditions on the Lie group the induced homeomorphism of the support is topologically equivalent to a translation of a compact group.read more
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Book
Introduction to Arithmetic Groups
TL;DR: A good introduction to the study of arithmetic subgroups of semisimple Lie groups can be found in this article, where the authors provide primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, and quasi-isometries.
Book
Ratner's Theorems on Unipotent Flows
TL;DR: In this paper, the authors present a collection of lecture notes aimed at graduate students, the first four chapters of "Ratner's Theorems on Unipotent Flows" can be read independently.
Book ChapterDOI
Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory
TL;DR: In this article, the authors present an exposition of homogeneous dynamics, that is, the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups.
Dynamics of Subgroup Actions on Homogeneous Spaces of Lie Groups and Applications to Number Theory
Dmitry Kleinbock,Nimish A. Shah +1 more
TL;DR: In this article, the authors present an exposition of homogeneous dynamics, that is, the dynamical and ergodic properties of actions on the homogeneous spaces of Lie groups.
Journal ArticleDOI
Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T)
Yehuda Shalom,Yehuda Shalom +1 more
TL;DR: In this paper, it was shown that finite invariant measures for k-algebraic actions are obtained only via actions of compact groups, which can be seen as a generalization of Kazhdan's property for algebraic groups.
References
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Linear algebraic groups
TL;DR: Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions are discussed in this paper.
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Theory of Lie Groups
TL;DR: Chevalley as mentioned in this paper introduced the notion of a Lie group as a global object in the calculus of exterior differential forms, and showed how to construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group).