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Article Contents

# Every action of a nonamenable group is the factor of a small action

• It is well known that if $G$ is a countable amenable group and $G ↷ (Y, \nu)$ factors onto $G ↷ (X, \mu)$, then the entropy of the first action must be at least the entropy of the second action. In particular, if $G ↷ (X, \mu)$ has infinite entropy, then the action $G ↷ (Y, \nu)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable nonamenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G ↷ (X, \mu)$ there is a $G$-invariant probability measure $\nu$ on $n^G$ such that $G ↷ (n^G, \nu)$ factors onto $G ↷ (X, \mu)$. For many nonamenable groups, $n$ can be chosen to be $4$ or smaller. We also obtain a similar result with respect to continuous actions on compact spaces and continuous factor maps.
Mathematics Subject Classification: Primary: 37A15; Secondary: 37A20, 37A35, 37B10, 37B40, 03E15.

 Citation:

•  [1] K. Ball, Factors of independent and identically distributed processes with non-amenable group actions, Ergodic Theory Dynam. Systems, 25 (2005), 711-730.doi: 10.1017/S0143385704001063. [2] L. Bowen, A measure-conjugacy invariant for free group actions, Ann. of Math. (2), 171 (2010), 1387-1400.doi: 10.4007/annals.2010.171.1387. [3] L. Bowen, Weak isomorphisms between Bernoulli shifts, Israel J. Math., 183 (2011), 93-102.doi: 10.1007/s11856-011-0043-3. [4] L. Bowen, Every countably infinite group is almost Ornstein, in Dynamical Systems and Group Actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 67-78.doi: 10.1090/conm/567/11234. [5] I. Epstein, Orbit inequivalent actions of non-amenable groups, preprint, arXiv:0707.4215. [6] D. Gaboriau, Coût des relations d'équivalance et des groupes, Invent. Math., 139 (2000), 41-98.doi: 10.1007/s002229900019. [7] D. Gaboriau and R. Lyons, A measurable-group-theoretic solution to von Neumann's problem, Invent. Math., 177 (2009), 533-540.doi: 10.1007/s00222-009-0187-5. [8] A. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.doi: 10.1007/978-1-4612-4190-4. [9] A. Kechris, Weak containment in the space of actions of a free group, Israel J. Math., 189 (2012), 461-507.doi: 10.1007/s11856-011-0182-6. [10] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, Amer. J. Math., 135 (2013), 721-761.doi: 10.1353/ajm.2013.0024. [11] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math., 48 (1987), 1-141.doi: 10.1007/BF02790325. [12] B. Seward, Burnside's problem, spanning trees, and tilings, to appear in Geometry & Topology. [13] J. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic, 18 (1980), 1-28.doi: 10.1016/0003-4843(80)90002-9. [14] A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302. [15] B. Weiss, Measurable dynamics, in Conference in Modern Analysis and Probability (eds. R. Beals, et. al.), Contemporary Mathematics, 26, American Mathematical Society, Providence, RI, 1984, 395-421.doi: 10.1090/conm/026/737417. [16] K. Whyte, Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture, Duke Math. J., 99 (1999), 93-112.doi: 10.1215/S0012-7094-99-09904-0.