Constructing infra-nilmanifolds admitting an Anosov diffeomorphism
type: Item
instance of: scholarly article
title: Constructing infra-nilmanifolds admitting an Anosov diffeomorphism
zbMATH Open document ID: 1252.37021
author: Karel Dekimpe
author: Kelly Verheyen
published in: Advances in Mathematics
publication date: 2011-12-02T00:00:00Z
review text: The authors study the existence of Anosov diffeomorphisms on infra-nilmanifols modeled on a free \(c\)-step nilpotent Lie group. They use the notion of a rational holonomy representation of infra-nilmanifolds which was introduced in [Groups Geom. Dyn. 3, No. 4, 555--578 (2009; Zbl 1185.37061)] by the same authors and which is a generalization of the notion of a rational holonomy representation of a flat manifold used by \textit{H. L. Porteous} [Topology 11, 307--315 (1972; Zbl 0237.58015)]. In the paper under review, the authors give an algebraic characterization of those infra-nilmanifolds modeled on a free \(c\)-step nilpotent Lie group and with abelian holonomy group admitting an Anosov diffeomorphism. They also prove that if an infra-nilmanifold \(M\) modeled on a free \(c\)-step nilpotent Lie group on \(n\) generators, with \(n\leq 2c+ 1\), admits an Anosov diffeomorphism then the holonomy group of \(M\) is abelian. The authors develop a new method for constructing examples of infra-nilmanifolds admitting Anosov diffeomorphisms. These examples include infra-nilmanifolds with cyclic holonomy groups and non-abelian holonomy groups.
reviewed by: Meera G. Mainkar
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 5985538
zbMATH Keywords: infra-nilmanifold
zbMATH Keywords: Anosov diffeomorphism
zbMATH Keywords: free nilpotent Lie group
MaRDI profile type: Publication
full work available at URL: j.aim.2011.08.008
OpenAlex ID: W2063313627
cites work: Nilmanifolds with Anosov Automorphism
cites work: Anosov automorphisms on compact nilmanifolds associated with graphs
cites work: Almost-Bieberbach groups: affine and polynomial structures
cites work: AFFINE STRUCTURES FOR CLOSED 3-DIMENSIONAL MANIFOLDS WITH NIL-GEOMETRY
cites work: Anosov diffeomorphisms on infra-nilmanifolds modeled on a free 2-step nilpotent Lie group
cites work: Anosov diffeomorphisms on nilmanifolds modelled on a free nilpotent Lie group
cites work: Q5494299
cites work: Q3342725
cites work: Examples of Anosov diffeomorphisms
cites work: Nilmanifolds of dimension ≤8 admitting Anosov diffeomorphisms
cites work: THERE ARE ONLY FINITELY MANY INFRA-NILMANIFOLDS UNDER EACH NILMANIFOLD
cites work: ANOSOV AUTOMORPHISMS ON CERTAIN CLASSES OF NILMANIFOLDS
cites work: Examples of Anosov Lie algebras
cites work: Anosov diffeomorphisms of nilmanifolds of dimension at most six
cites work: There are No New Anosov Diffeomorphisms on Tori
cites work: Q4935965
cites work: Anosov automorphisms of nilpotent Lie algebras
cites work: Anosov diffeomorphisms of flat manifolds
cites work: Q3663506
cites work: Q3813882
DOI: 10.1016/J.AIM.2011.08.008
description: scientific article; zbMATH DE number 5985538