# American Institute of Mathematical Sciences

2014, 8(2): 251-270. doi: 10.3934/jmd.2014.8.251

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

 1 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, United States

Received  December 2013 Published  November 2014

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.
Citation: Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251
##### References:
 [1] K. Ball, Factors of independent and identically distributed processes with non-amenable group actions,, Ergodic Theory Dynam. Systems, 25 (2005), 711. doi: 10.1017/S0143385704001063. [2] L. Bowen, A measure-conjugacy invariant for free group actions,, Ann. of Math. (2), 171 (2010), 1387. doi: 10.4007/annals.2010.171.1387. [3] L. Bowen, Weak isomorphisms between Bernoulli shifts,, Israel J. Math., 183 (2011), 93. doi: 10.1007/s11856-011-0043-3. [4] L. Bowen, Every countably infinite group is almost Ornstein,, in Dynamical Systems and Group Actions, (2012), 67. doi: 10.1090/conm/567/11234. [5] I. Epstein, Orbit inequivalent actions of non-amenable groups,, preprint, (). [6] D. Gaboriau, Coût des relations d'équivalance et des groupes,, Invent. Math., 139 (2000), 41. 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. doi: 10.1007/s00222-009-0187-5. [8] A. Kechris, Classical Descriptive Set Theory,, Graduate Texts in Mathematics, (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. doi: 10.1007/s11856-011-0182-6. [10] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy,, Amer. J. Math., 135 (2013), 721. 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. 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. doi: 10.1016/0003-4843(80)90002-9. [14] A. M. Stepin, Bernoulli shifts on groups,, Dokl. Akad. Nauk SSSR, 223 (1975), 300. [15] B. Weiss, Measurable dynamics,, in Conference in Modern Analysis and Probability (eds. R. Beals, (1984), 395. doi: 10.1090/conm/026/737417. [16] K. Whyte, Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture,, Duke Math. J., 99 (1999), 93. doi: 10.1215/S0012-7094-99-09904-0.

show all references

##### References:
 [1] K. Ball, Factors of independent and identically distributed processes with non-amenable group actions,, Ergodic Theory Dynam. Systems, 25 (2005), 711. doi: 10.1017/S0143385704001063. [2] L. Bowen, A measure-conjugacy invariant for free group actions,, Ann. of Math. (2), 171 (2010), 1387. doi: 10.4007/annals.2010.171.1387. [3] L. Bowen, Weak isomorphisms between Bernoulli shifts,, Israel J. Math., 183 (2011), 93. doi: 10.1007/s11856-011-0043-3. [4] L. Bowen, Every countably infinite group is almost Ornstein,, in Dynamical Systems and Group Actions, (2012), 67. doi: 10.1090/conm/567/11234. [5] I. Epstein, Orbit inequivalent actions of non-amenable groups,, preprint, (). [6] D. Gaboriau, Coût des relations d'équivalance et des groupes,, Invent. Math., 139 (2000), 41. 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. doi: 10.1007/s00222-009-0187-5. [8] A. Kechris, Classical Descriptive Set Theory,, Graduate Texts in Mathematics, (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. doi: 10.1007/s11856-011-0182-6. [10] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy,, Amer. J. Math., 135 (2013), 721. 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. 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. doi: 10.1016/0003-4843(80)90002-9. [14] A. M. Stepin, Bernoulli shifts on groups,, Dokl. Akad. Nauk SSSR, 223 (1975), 300. [15] B. Weiss, Measurable dynamics,, in Conference in Modern Analysis and Probability (eds. R. Beals, (1984), 395. doi: 10.1090/conm/026/737417. [16] K. Whyte, Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture,, Duke Math. J., 99 (1999), 93. doi: 10.1215/S0012-7094-99-09904-0.
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