Arithmeticity and topology of smooth actions of higher rank abelian groups
Anatole Katok - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email) 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 on its factor by $\pm\mathrm{Id}$. 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.
Keywords: Actions of higher rank abelian groups, maximal rank actions, Lyapunov
exponents, geometric structure, arithmetic structure, entropy, rigidity.
Received: November 2014; Revised: April 2016; Available Online: May 2016. |