Arithmeticity and topology of smooth actions of higher rank abelian groups

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Abstract

We prove that any smooth action of ℤm-1, m ≥ 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 on its factor by ±Id. Furthermore this isomorphism has nice geometric properties; in particular, it is smooth in the sense of Whitney on a set whose complement has arbitrarily small measure. We further derive restrictions on topology of manifolds that may admit such actions, for example, excluding spheres and obtaining lower estimate on the first Betti number in the odd-dimensional case.

Original languageEnglish (US)
Pages (from-to)135-172
Number of pages38
JournalJournal of Modern Dynamics
Volume10
DOIs
StatePublished - Jan 1 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

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