Nilmanifolds with Anosov automorphism

Abstract

In his survey article [14] S Smalc raised the problem of classifying all Anosov automorphisms of compact manifolds. His conjecture, which is now supported by results of J- Franks [5J and A, Manning [91, is that any Anosov automorphism of a compact manifold is topologically conjugate to a hyperbolic infra-nilmanifold automorphism. This highlights another problem, that of classifying all nilmanifolds which admit Anosov automorphisms The question was first raised in a somewhat weaker form by Anosov, in the Moscow congress. In (14J Smale described an example, due to Borel, of a non-toral nilmanifold admitting Anosov automorphisms. Sub-sequently L. Auslander and J. Scheuneman [1] introduced another class of nilmani-folds admitting Anosov automorphisms. On the other hand it is easy to construct examples of nilmanifolds which do not admit Anosov automorphisms. In this note, using some results about algebraic groups and arithmetic sub-groups, we propose a general construction of nilmanifolds with Anosov automorphisms. Apart from providing a more general perspective (cf. Corollary 2.3 and the introduction to 3) we are also able to construct some new examples (cf. Corollary 3.2). It is conceivable that the method may lead to a complete classification of nilmanifolds with Anosov automorphisms and the author hopes to return to the subject.