Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds.
type: Item
instance of: scholarly article
title: Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds.
zbMATH Open document ID: 1075.17007
DOI: 10.1016/S0166-8641(02)00224-9
author: Karel Dekimpe
author: Kyung Bai Lee
published in: Topology and its Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(L\) be a finite dimensional real Lie algebra. An automorphism of \(L\) is called expanding (hyperbolic) if all of its eigenvalues are outside the unit circle (if it has eigenvalues both inside and outside the unit circle, but none on the unit circle). \(L\) admits an expanding (hyperbolic) automorphism if and only if \(L\) admits a positive (mixed, with the zero part equal to zero) \(R\)-grading. It is then shown that any infra-nilmanifold which admits an expanding map or an Anosov diffeomorphism has a complete affine structure.
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 37D20
zbMATH DE Number: 1956726
zbMATH Keywords: hyperbolic/expanding automorphism
reviewed by: Ernest L. Stitzinger
MaRDI profile type: Publication
cites work: Holonomy of flat affinely connected manifolds
cites work: Almost-Bieberbach groups: affine and polynomial structures
cites work: Affine structures on 4-step nilpotent Lie algebras
cites work: Expanding maps on infra-nilmanifolds of homogeneous type
cites work: Affine structures on a class of virtually nilpotent groups
cites work: Derivations of Nilpotent Lie Algebras
cites work: Expanding endomorphisms of flat manifolds
cites work: Q5608683
cites work: Affine manifolds with nilpotent holonomy
cites work: Three-dimensional affine crystallographic groups
cites work: Groups of polynomial growth and expanding maps. Appendix by Jacques Tits
cites work: Q5628968
cites work: Anosov maps, polycyclic groups and homology
cites work: Homogeneous Lie Algebras and Expanding Automorphisms
cites work: Complete left-invariant affine structures on nilpotent Lie groups
cites work: Expanding maps on 2-step infra-nilmanifolds
cites work: Q4762707
cites work: There are No New Anosov Diffeomorphisms on Tori
cites work: On fundamental groups of complete affinely flat manifolds
cites work: Factor spaces of solvable groups
cites work: Affine Structures on Three-Step Nilpotent Lie Algebras
cites work: The structure of complete left-symmetric algebras
cites work: Q5615933
full work available at URL: s0166-8641(02)00224-9
OpenAlex ID: W2027256591
description: scientific article; zbMATH DE number 1956726