Some two-step and three-step nilpotent Lie groups with small automorphism groups. (English) Zbl 1023.22006
Let \(G\) be a connected Lie group and \(\operatorname{Aut}(G)\) the group of all automorphisms of \(G\). The paper studies the action of \(\operatorname{Aut}(G)\) on \(G\), especially topological properties of its orbits. For instance, when does the action of \(\operatorname{Aut}(G)\) on \(G\) have a dense orbit? If this holds, then \(G\) is nilpotent [cf. the author’s article in: Ergodic theory and harmonic analysis (Mumbai, 1999), Sankhya, Ser. A 62, 360-366 (2000; Zbl 0986.22003)]. However, it does not hold for all nilpotent Lie groups. When \(G\) is a vector space, \(\operatorname{Aut}(G)\) is its general linear group and the action has an open dense orbit, namely the complement of the zero.
The author here constructs, making use of a representation of \(\text{PSL}(2,\mathbb{R})\), an example of a two-step simply connected nilpotent Lie group \(G\) for which the action of \(\operatorname{Aut}(G)\) has no dense orbit. This enables him to obtain new examples of compact nilmanifolds which do not admit Anosov automorphisms.
He also gives examples of three-step simply connected nilpotent Lie groups \(G\) such that \(\operatorname{Aut}(G)\) is nilpotent and acts on the Lie algebra of \(G\) by unipotent transformations, what implies in particular that all orbits of the action are closed.
The author here constructs, making use of a representation of \(\text{PSL}(2,\mathbb{R})\), an example of a two-step simply connected nilpotent Lie group \(G\) for which the action of \(\operatorname{Aut}(G)\) has no dense orbit. This enables him to obtain new examples of compact nilmanifolds which do not admit Anosov automorphisms.
He also gives examples of three-step simply connected nilpotent Lie groups \(G\) such that \(\operatorname{Aut}(G)\) is nilpotent and acts on the Lie algebra of \(G\) by unipotent transformations, what implies in particular that all orbits of the action are closed.
Reviewer: Hidenori Fujiwara (Iizuka)
MSC:
| 22D45 | Automorphism groups of locally compact groups |
| 22E25 | Nilpotent and solvable Lie groups |
| 22D40 | Ergodic theory on groups |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
Citations:
Zbl 0986.22003References:
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