Superrigidity of lattices in solvable Lie groups
type: Item
instance of: scholarly article
title: Superrigidity of lattices in solvable Lie groups
zbMATH Open document ID: 0844.22015
DOI: 10.1007/BF01231442
published in: Inventiones Mathematicae
publication date: 1996-03-31T00:00:00Z
full work available at URL: https://eudml.org/doc/144318
review text: Let \(\Gamma\) be a closed cocompact subgroup of a simply connected solvable Lie group \(G\), such that \(Adj_G \Gamma\) has the same Zariski closure as \(Ad G\). Main results: Any finite-dimensional representation \(\alpha : \Gamma \to GL_n (\mathbb{R})\) can be virtually extended to a representation of \(G\). If \(\Gamma\) is isomorphic to a closed cocompact subgroup \(\Gamma'\) of another simply connected cocompact Lie group \(G'\), then any isomorphism from \(\Gamma\) to \(\Gamma'\) extends to a crossed isomorphism from \(G\) to \(G'\). -- The author uses two technical tools -- the nilshadow map and the existence of syndetic hulls. The main superrigidity theorem is described for solvable groups and for lattices in nonsolvable groups.
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 861961
zbMATH Keywords: simply connected solvable Lie group
zbMATH Keywords: Zariski closure
zbMATH Keywords: representation
zbMATH Keywords: crossed isomorphism
zbMATH Keywords: nilshadow map
zbMATH Keywords: syndetic hulls
zbMATH Keywords: superrigidity theorem
author: David Witte Morris
reviewed by: Bohumil František Šmarda
MaRDI profile type: Publication
Wikidata QID: Q115393815
cites work: An exposition of the structure of solvmanifolds. Part I: Algebraic theory
cites work: Splitting theorems and the structure of solvmanifolds
cites work: Density properties for certain subgroups of semi-simple groups without compact components
cites work: Q4081425
cites work: On ergodic quasi-invariant measures of group automorphism
cites work: Three-dimensional affine crystallographic groups
cites work: LATTICES IN SOLVABLE LIE GROUPS AND DEFORMATIONS OF HOMOGENEOUS SPACES
cites work: GENERALIZED LJAPUNOV THEOREM ON MAL'CEV MANIFOLDS
cites work: Q3755676
cites work: Q3897665
cites work: Q5339349
cites work: On some types of topological groups
cites work: Über nilpotente topologische Gruppen, I
cites work: Q5732712
cites work: Analytic representations of polycyclic groups
cites work: Zariski density in Lie groups
cites work: Factor spaces of solvable groups
cites work: Arithmetic subgroups of groups with radical
cites work: Q3244133
cites work: Rigidity problem for lattices in solvable Lie groups
cites work: Q4527707
cites work: On density properties of certain subgroups of locally compact groups
cites work: Zero-Entropy Affine Maps on Homogeneous Spaces
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description: scientific article; zbMATH DE number 861961