Arithmeticity and topology of smooth actions of higher rank abelian groups (Q316989)

Let \(\alpha\) be a smooth action of \(\mathbb Z^{m-1}\), \(m\geq 3\), on an \(m\)-dimensional manifold \(M\) (not necessarily compact) that preserves a measure \(\mu\), such that all non-identity elements of the suspension have positive entropy.NEWLINENEWLINEThe authors prove that \(\alpha\) is essentially algebraic, that is, \(\alpha\) is isomorphic (up to a finite permutation) to an affine action on the torus or on its factor by \(\pm \mathrm{Id}\). This isomorphism is smooth, in the sense of Whitney, on a set whose complement has arbitrarily small measure.NEWLINENEWLINEThe authors find some restrictions on the topology of manifolds that may admit this type of actions.