Arithmeticity and topology of smooth actions of higher rank abelian groups

TL;DR

This paper shows that smooth higher-rank abelian group actions with positive entropy are essentially algebraic, leading to restrictions on the topology of manifolds that can admit such actions.

Contribution

It establishes a rigidity result for smooth actions of higher-rank abelian groups, proving they are algebraic up to finite permutation and analyzing their geometric properties.

Findings

Actions are isomorphic to affine torus actions or their factors.

Restrictions on manifold topology, excluding spheres.

Provides lower bounds on the first Betti number in odd dimensions.

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.