Arithmeticity and topology of smooth actions of higher rank abelian groups

Abstract

We prove that any smooth action of $\mathbb Z^{m-1}, m\ge 3$ on an $m$-dimensional manifold that preserves a measure such that all non-identity elements of the suspension have positive entropy is essentially algebraic, i.e. isomorphic up to a finite permutation to an affine action on the torus or its factor by $\pm\Id$. Furthermore this isomorphism has nice geometric properties, in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrary small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining below estimate on the first Betti number in the odd-dimensional case.