Some two-step and three-step nilpotent Lie groups with small automorphism groups

Abstract

We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are `small' in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we get also new examples of compact manifolds covered by two-step simply connected Lie groups, which do not admit Anosov automorphisms.