TY - JOUR
T1 - Arithmeticity and topology of smooth actions of higher rank abelian groups
AU - Katok, Anatole
AU - Hertz, Federico Rodriguez
N1 - Funding Information:
AK: Research supported by NSF grants DMS 1002554 and DMS 1304830. FRH: Research supported by NSF grants DMS 1201326 and DMS 1500947.
Publisher Copyright:
© 2016 AIMSCIENCES.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84966349559&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84966349559&partnerID=8YFLogxK
U2 - 10.3934/jmd.2016.10.135
DO - 10.3934/jmd.2016.10.135
M3 - Article
AN - SCOPUS:84966349559
SN - 1930-5311
VL - 10
SP - 135
EP - 172
JO - Journal of Modern Dynamics
JF - Journal of Modern Dynamics
ER -