Abstract
We consider Anosov actions of a Lie group G of dimension k on a closed manifold of dimension \(k+n\). We introduce the notion of Nil-Anosov action of G (which includes the case where G is nilpotent) and establishes the invariance by the entire group G of the associated stable and unstable foliations. We then prove a spectral decomposition Theorem for such an action when the group G is nilpotent. Finally, we focus on the case where G is nilpotent and the unstable bundle has codimension one. We prove that in this case the action is a Nil-extension over an Anosov action of an abelian Lie group. In particular: if \(n \ge 3,\) then the action is topologically transitive, if \(n=2,\) then the action is a Nil-extension over an Anosov flow.
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