Constructing infra-nilmanifolds admitting an Anosov diffeomorphism
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.
scientific article; zbMATH DE number 5985538
scientific article