# American Institute of Mathematical Sciences

2019, 15: 1-39. doi: 10.3934/jmd.2019012

## Krieger's finite generator theorem for actions of countable groups Ⅱ

 Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Received  May 23, 2018 Revised  November 2018 Published  February 2019

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért–Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (ⅰ) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ⅱ) Bernoulli shifts have completely positive Rokhlin entropy; and (ⅲ) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.

Citation: Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012
##### References:
 [1] M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action, Ergodic Theory and Dynamical Systems, 23 (2013), 323-333. doi: 10.1017/S0143385711000988. [2] A. Alpeev and B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅲ, preprint, arXiv: 1705.09707. [3] P. Ara, K. C. O'Meara and F. Perera, Stable finiteness of group rings in arbitrary characteristic, Advances in Math., 170 (2002), 224-238. doi: 10.1006/aima.2002.2075. [4] L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, Journal of the American Mathematical Society, 23 (2010), 217-245. doi: 10.1090/S0894-0347-09-00637-7. [5] L. Bowen, Sofic entropy and amenable groups, Ergod. Th. & Dynam. Sys., 32 (2012), 427-466. doi: 10.1017/S0143385711000253. [6] 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. [7] M. Burger and A. Valette, Idempotents in complex group rings: Theorems of Zalesskii and Bass revisited, Journal of Lie Theory, 8 (1998), 219-228. [8] V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, With an appendix by Vladimir Pestov, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. doi: 10.1007/978-3-319-19333-5. [9] N.-P. Chung, Topological pressure and the variational principle for actions of sofic groups, Ergodic Theory and Dynamical Systems, 33 (2013), 1363-1390. doi: 10.1017/S0143385712000429. [10] T. Downarowicz, Entropy in Dynamical Systems, Cambridge University Press, New York, 2011. doi: 10.1017/CBO9780511976155. [11] G. Elek and E. Szabó, Sofic groups and direct finiteness, Journal of Algebra, 280 (2004), 426-434. doi: 10.1016/j.jalgebra.2004.06.023. [12] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [13] W. Gottschalk, Some general dynamical notions, in Recent Advances in Topological Dynamics, Lecture Notes in Mathematics, 318, Springer, Berlin, 1973, 120–125. [14] C. Grillenberger and U. Krengel, On marginal distributions and isomorphisms of stationary processes, Math. Z., 149 (1976), 131-154. doi: 10.1007/BF01301571. [15] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. European Math. Soc., 1 (1999), 109-197. doi: 10.1007/PL00011162. [16] I. Kaplansky, Fields and Rings, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, IL, 1972. [17] A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4. [18] A. Kechris, S. Solecki and S. Todorcevic, Borel chromatic numbers, Adv. in Math., 141 (1999), 1-44. doi: 10.1006/aima.1998.1771. [19] D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632. doi: 10.4171/GGD/200. [20] D. Kerr, Bernoulli actions of sofic groups have completely positive entropy, Israel Journal of Math., 202 (2014), 461-474. doi: 10.1007/s11856-014-1077-0. [21] D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558. doi: 10.1007/s00222-011-0324-9. [22] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, American Journal of Mathematics, 135 (2013), 721-761. doi: 10.1353/ajm.2013.0024. [23] D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672. doi: 10.4171/GGD/142. [24] A. N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces, Doklady of Russian Academy of Sciences, 119 (1958), 861-864. [25] A. N. Kolmogorov, Entropy per unit time as a metric invariant for automorphisms, Doklady of Russian Academy of Sciences, 124 (1959), 754-755. [26] D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-352. doi: 10.1016/0001-8708(70)90029-0. [27] D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math., 5 (1970), 339-348. doi: 10.1016/0001-8708(70)90008-3. [28] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d'Analyse Mathématique, 48 (1987), 1-141. doi: 10.1007/BF02790325. [29] D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Annals of Mathematics, 151 (2000), 1119-1150. doi: 10.2307/121130. [30] Ya. G. Sinam${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771. [31] B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅰ, Inventiones Mathematicae, 215 (2019), 265-310. doi: 10.1007/s00222-018-0826-9. [32] B. Seward, Weak containment, Pinsker algebras, and Rokhlin entropy, preprint, arXiv: 1602.06680. [33] B. Seward and R. D. Tucker-Drob, Borel structurability on the 2-shift of a countable group, Ann. Pure Appl. Logic, 167 (2016), 1-21. doi: 10.1016/j.apal.2015.07.005. [34] A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302. [35] B. Weiss, Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350–359.

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##### References:
 [1] M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action, Ergodic Theory and Dynamical Systems, 23 (2013), 323-333. doi: 10.1017/S0143385711000988. [2] A. Alpeev and B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅲ, preprint, arXiv: 1705.09707. [3] P. Ara, K. C. O'Meara and F. Perera, Stable finiteness of group rings in arbitrary characteristic, Advances in Math., 170 (2002), 224-238. doi: 10.1006/aima.2002.2075. [4] L. Bowen, Measure conjugacy invariants for actions of countable sofic groups, Journal of the American Mathematical Society, 23 (2010), 217-245. doi: 10.1090/S0894-0347-09-00637-7. [5] L. Bowen, Sofic entropy and amenable groups, Ergod. Th. & Dynam. Sys., 32 (2012), 427-466. doi: 10.1017/S0143385711000253. [6] 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. [7] M. Burger and A. Valette, Idempotents in complex group rings: Theorems of Zalesskii and Bass revisited, Journal of Lie Theory, 8 (1998), 219-228. [8] V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, With an appendix by Vladimir Pestov, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. doi: 10.1007/978-3-319-19333-5. [9] N.-P. Chung, Topological pressure and the variational principle for actions of sofic groups, Ergodic Theory and Dynamical Systems, 33 (2013), 1363-1390. doi: 10.1017/S0143385712000429. [10] T. Downarowicz, Entropy in Dynamical Systems, Cambridge University Press, New York, 2011. doi: 10.1017/CBO9780511976155. [11] G. Elek and E. Szabó, Sofic groups and direct finiteness, Journal of Algebra, 280 (2004), 426-434. doi: 10.1016/j.jalgebra.2004.06.023. [12] E. Glasner, Ergodic Theory Via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101. [13] W. Gottschalk, Some general dynamical notions, in Recent Advances in Topological Dynamics, Lecture Notes in Mathematics, 318, Springer, Berlin, 1973, 120–125. [14] C. Grillenberger and U. Krengel, On marginal distributions and isomorphisms of stationary processes, Math. Z., 149 (1976), 131-154. doi: 10.1007/BF01301571. [15] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. European Math. Soc., 1 (1999), 109-197. doi: 10.1007/PL00011162. [16] I. Kaplansky, Fields and Rings, Chicago Lectures in Mathematics, The University of Chicago Press, Chicago, IL, 1972. [17] A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4. [18] A. Kechris, S. Solecki and S. Todorcevic, Borel chromatic numbers, Adv. in Math., 141 (1999), 1-44. doi: 10.1006/aima.1998.1771. [19] D. Kerr, Sofic measure entropy via finite partitions, Groups Geom. Dyn., 7 (2013), 617-632. doi: 10.4171/GGD/200. [20] D. Kerr, Bernoulli actions of sofic groups have completely positive entropy, Israel Journal of Math., 202 (2014), 461-474. doi: 10.1007/s11856-014-1077-0. [21] D. Kerr and H. Li, Entropy and the variational principle for actions of sofic groups, Invent. Math., 186 (2011), 501-558. doi: 10.1007/s00222-011-0324-9. [22] D. Kerr and H. Li, Soficity, amenability, and dynamical entropy, American Journal of Mathematics, 135 (2013), 721-761. doi: 10.1353/ajm.2013.0024. [23] D. Kerr and H. Li, Bernoulli actions and infinite entropy, Groups Geom. Dyn., 5 (2011), 663-672. doi: 10.4171/GGD/142. [24] A. N. Kolmogorov, New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces, Doklady of Russian Academy of Sciences, 119 (1958), 861-864. [25] A. N. Kolmogorov, Entropy per unit time as a metric invariant for automorphisms, Doklady of Russian Academy of Sciences, 124 (1959), 754-755. [26] D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math., 4 (1970), 337-352. doi: 10.1016/0001-8708(70)90029-0. [27] D. Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math., 5 (1970), 339-348. doi: 10.1016/0001-8708(70)90008-3. [28] D. Ornstein and B. Weiss, Entropy and isomorphism theorems for actions of amenable groups, Journal d'Analyse Mathématique, 48 (1987), 1-141. doi: 10.1007/BF02790325. [29] D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Annals of Mathematics, 151 (2000), 1119-1150. doi: 10.2307/121130. [30] Ya. G. Sinam${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk SSSR, 124 (1959), 768-771. [31] B. Seward, Krieger's finite generator theorem for actions of countable groups Ⅰ, Inventiones Mathematicae, 215 (2019), 265-310. doi: 10.1007/s00222-018-0826-9. [32] B. Seward, Weak containment, Pinsker algebras, and Rokhlin entropy, preprint, arXiv: 1602.06680. [33] B. Seward and R. D. Tucker-Drob, Borel structurability on the 2-shift of a countable group, Ann. Pure Appl. Logic, 167 (2016), 1-21. doi: 10.1016/j.apal.2015.07.005. [34] A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR, 223 (1975), 300-302. [35] B. Weiss, Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350–359.
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