Abstract
We prove that every Bernoulli action of a sofic group has completely positive entropy with respect to every sofic approximation net. We also prove that every Bernoulli action of a finitely generated free group has the property that each of its nontrivial factors with a finite generating partition has positive f-invariant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Ball, Factors of independent and identically distributed processes with non-amenable group actions, Ergodic Theory and Dynamical Systems 25 (2005), 711–730.
L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, Journal of the American Mathematical Society 23 (2010), 217–245.
L. Bowen, The ergodic theory of free group actions: entropy and the f-invariant, Groups, Geometry, and Dynamics 4 (2010), 419–432.
L. Bowen, A measure-conjugacy invariant for free group actions, Annals of Mathematics 171 (2010), 1387–1400.
L. Bowen, Weak isomorphisms between Bernoulli shifts, Israel Journal of Mathematics 183 (2011), 93–102.
E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, Vol. 101, American Mathematical Society, Providence, RI, 2003.
M. Gromov and V. D. Milman, A topological application of the isoperimetric inequality, American Journal of Mathematics 105 (1983), 843–854.
D. Kerr, Sofic measure entropy via finite partitions, Groups, Geometry, and Dynamics 7 (2013), 617–632.
D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups, Geometry, and Dynamics 5 (2011), 663–672.
D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Inventiones Mathematicae 186 (2011), 501–558.
B. Maurey, Construction de suites symétriques, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences 288 (1979), 679–681.
A. Nica, Asymptotically free families of random unitaries in symmetric groups, Pacfic Journal of Mathematics 157 (1993), 295–310.
D. S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Mathematics 10 (1973), 49–62.
D. S. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d’Analyse Mathématique 48 (1987), 1–141.
S. Popa, Some computations of 1-cohomology groups and construction of non orbit equivalent actions, Journal of the Institute of of Mathematics of Jussieu 5 (2006), 309–332.
S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups, Inventiones Mathematicae 170 (2007), 243–295.
S. Popa, On the superrigidity of malleable actions with spectral gap, Journal of the American Mathematical Society 21 (2008), 981–1000.
S. Popa and R. Sasyk, On the cohomology of Bernoulli actions, Ergodic Theory and Dynamical Systems 27 (2007), 241–251.
D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Annals of Mathematics 151 (2000), 1119–1150.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kerr, D. Bernoulli actions of sofic groups have completely positive entropy. Isr. J. Math. 202, 461–474 (2014). https://doi.org/10.1007/s11856-014-1077-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-014-1077-0