Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T09:20:05.682Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic models and Bernoullicity

Part of: Locally compact groups and their algebras Ergodic theory

Published online by Cambridge University Press:  19 August 2021

Ben Hayes
Ben Hayes*
Affiliation:
Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, Charlottesville, VA22904, USAbrh5c@virginia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

Keywords

Harmonic modelalgebraic actionsBernoulli shift

MSC classification

Primary: 37A15: General groups of measure-preserving transformations 37A35: Entropy and other invariants, isomorphism, classification 37A55: Relations with the theory of $C^*$-algebras 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 10 , October 2021 , pp. 2160 - 2198
Check for updates
Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author gratefully acknowledges support from NSF Grants DMS-1827376 and DMS-2000105.

References

Alexopoulos, G. K., Random walks on discrete groups of polynomial volume growth, Ann. Probab. 30 (2002), 723–801; MR 1905856.CrossRefGoogle Scholar
Alpeev, A., Meyerovitch, T. and Ryu, S., Predictability, topological entropy, and invariant random orders, Proc. Amer. Math. Soc. 149 (2021), 1443–1457; MR 4242303.CrossRefGoogle Scholar
Austin, T., The geometry of model spaces for probability-preserving actions of sofic groups, Anal. Geom. Metr. Spaces 4 (2016), 160–186; MR 3543677.Google Scholar
Bekka, B., de la Harpe, P. and Valette, A., Kazhdan's property ( T ), New Mathematical Monographs, vol. 11 (Cambridge University Press, Cambridge, 2008); MR 2415834.CrossRefGoogle Scholar
Bowen, L., Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc. 23 (2010), 217–245; MR 2552252.CrossRefGoogle Scholar
Bowen, L., Entropy for expansive algebraic actions of residually finite groups, Ergodic Theory Dynam. Systems 31 (2011), 703–718; MR 2794944.CrossRefGoogle Scholar
Bowen, L., Every countably infinite group is almost Ornstein, in Dynamical systems and group actions, Contemporary Mathematics, vol. 567, eds L. Bowen, R. Grigorchuk and Y. Vorobets (American Mathematical Society, Providence, RI, 2012), 67–78.CrossRefGoogle Scholar
Bowen, L., Zero entropy is generic, Entropy 18 (2016), Paper No. 220, 20; MR 3530042.CrossRefGoogle Scholar
Bowen, L., Finitary random interlacements and the Gaboriau-Lyons problem, Geom. Funct. Anal. 29 (2019), 659–689; MR 3962876.CrossRefGoogle Scholar
Bowen, L., Examples in the entropy theory of countable group actions, Ergodic Theory Dynam. Systems 40 (2020), 2593–2680; MR 4138907.CrossRefGoogle Scholar
Bowen, L. and Li, H., Harmonic models and spanning forests of residually finite groups, J. Funct. Anal. 263 (2012), 1769–1808; MR 2956925.CrossRefGoogle Scholar
Brown, L. G., Lidskiĭ's theorem in the type II case, in Geometric methods in operator algebras (Kyoto, 1983), Pitman Research Notes in Mathematics Series, vol. 123, eds H. Araki and E. Effros (Longman, Harlow, 1986), 1–35; MR 866489.Google Scholar
Cannon, J. W., Floyd, W. J. and Parry, W. R., Introductory notes on Richard Thompson's groups, Enseign. Math. (2) 42 (1996), 215–256; MR 1426438.Google Scholar
Chifan, I. and Ioana, A., Ergodic subequivalence relations induced by a Bernoulli action, Geom. Funct. Anal. 20 (2010), 53–67; MR 2647134.CrossRefGoogle Scholar
Conway, J. B., A course in functional analysis, second edition, Graduate Texts in Mathematics, vol. 96 (Springer, New York, 1990); MR 1070713.Google Scholar
Dehornoy, P., Braid groups and left distributive operations, Trans. Amer. Math. Soc. 345 (1994), 115–150; MR 1214782.CrossRefGoogle Scholar
Deninger, C., Fuglede-Kadison determinants and entropy for actions of discrete amenable groups, J. Amer. Math. Soc. 19 (2006), 737–758; MR 2220105.CrossRefGoogle Scholar
Deninger, C. and Schmidt, K., Expansive algebraic actions of discrete residually finite amenable groups and their entropy, Ergodic Theory Dynam. Systems 27 (2007), 769–786.CrossRefGoogle Scholar
Dixmier, J., Formes linéaires sur un anneau d'opérateurs, Bull. Soc. Math. France 81 (1953), 9–39; MR 59485.CrossRefGoogle Scholar
da Silva, R. C., Lecture notes on noncommutative lp-spaces, Preprint (2018), arXiv:1803.02390.Google Scholar
Gaboriau, D. and Lyons, R., A measurable-group-theoretic solution to von Neumann's problem, Invent. Math. 177 (2009), 533–540; MR 2534099.CrossRefGoogle Scholar
Gaboriau, D. and Seward, B., Factors of Bernoulli and treeability, Preprint (2019).Google Scholar
Ghys, E., Groups acting on the circle, Enseign. Math. (2) 47 (2001), 329–407; MR 1876932.Google Scholar
Grigorchuk, R. I., Degrees of growth of $p$-groups and torsion-free groups, Mat. Sb. (N.S.) 126 (1985), 194–214, 286; MR 784354.Google Scholar
Grigorchuk, R. I. and Machí, A., On a group of intermediate growth that acts on a line by homeomorphisms, Mat. Zametki 53 (1993), 46–63; MR 1220809.Google Scholar
Hayes, B., Fuglede–Kadison determinants and sofic entropy, Geom. Funct. Anal. 26 (2016), 520–606; MR 3513879.CrossRefGoogle Scholar
Hayes, B., Independence tuples and Deninger's problem, Groups Geom. Dyn. 11 (2017), 245–289; MR 3641841.CrossRefGoogle Scholar
Hayes, B., Max-min theorems for weak containment, square summable homoclinic points, and completely positive entropy, Indiana Univ. Math. J., to appear. Preprint (2019).Google Scholar
Hayes, B., Weak equivalence to Bernoulli shifts for some algebraic actions, Proc. Amer. Math. Soc. 147 (2019), 2021–2032; MR 3937679.CrossRefGoogle Scholar
Hayes, B., Relative entropy and the Pinsker product formula for sofic groups, Groups Geom. Dyn., to appear. Preprint (2021).Google Scholar
Houdayer, C., Invariant percolation and measured theory of nonamenable groups [after Gaboriau-Lyons, Ioana, Epstein], in Séminaire Bourbaki, Vol. 2010/2011, Exposés 1027–1042, Astérisque, No. 348 (Société Mathématique de France, 2012), Exp. No. 1039, ix, 339–374; MR 3051202.Google Scholar
Hyde, J. and Lodha, Y., Finitely generated infinite simple groups of homeomorphisms of the real line, Invent. Math. 218 (2019), 83–112; MR 3994586.Google Scholar
Katznelson, Y., Ergodic automorphisms of $T^n$ are Bernoulli shifts, Israel J. Math. 10 (1971), 186–195; MR 0294602.CrossRefGoogle Scholar
Kerr, D., Bernoulli actions of sofic groups have completely positive entropy, Israel J. Math. 202 (2014), 461–474; MR 3265329.CrossRefGoogle Scholar
Kerr, D. and Li, H., Entropy and the variational principle for actions of sofic groups, Invent. Math. 186 (2011), 501–558; MR 2854085.CrossRefGoogle Scholar
Kim, S. H., Koberda, T. and Lodha, Y., Chain groups of homeomorphisms of the interval, Ann. Sci. Éc. Norm. Supér (4), to appear. Preprint (2019).Google Scholar
Kitchens, B. and Schmidt, K., Automorphisms of compact groups, Ergodic Theory Dynam. Systems 9 (1989), 691–735; MR 1036904.CrossRefGoogle Scholar
Li, H., Sofic mean dimension, Adv. Math. 244 (2014), 570–604.CrossRefGoogle Scholar
Li, H. and Thom, A., Entropy, determinants, and $L^2$-torsion, J. Amer. Math. Soc. 27 (2014), 239–292; MR 3110799.CrossRefGoogle Scholar
Lind, D. A., Ergodic automorphisms of the infinite torus are Bernoulli, Israel J. Math. 17 (1974), 162–168; MR 0346130.CrossRefGoogle Scholar
Lind, D. and Schmidt, K., New examples of Bernoulli algebraic actions, Preprint (2021), arXiv:1905.09966.Google Scholar
Lind, D., Schmidt, K. and Verbitskiy, E., Entropy and growth rate of periodic points of algebraic $\Bbb Z^d$-actions, in Dynamical numbers—interplay between dynamical systems and number theory, Contemporary Mathematics, vol. 532, eds S. Kolyada et al. (American Mathematical Society, Providence, RI, 2010), 195–211; MR 2762141.CrossRefGoogle Scholar
Lind, D., Schmidt, K. and Verbitskiy, E., Homoclinic points, atoral polynomials, and periodic points of algebraic $\Bbb Z^d$-actions, Ergodic Theory Dynam. Systems 33 (2013), 1060–1081; MR 3082539.CrossRefGoogle Scholar
Lind, D., Schmidt, K. and Ward, T., Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), 593–629; MR 1062797.CrossRefGoogle Scholar
Martineau, S., Ergodicity and indistinguishability in percolation theory, Enseign. Math. 61 (2015), 285–319; MR 3539840.CrossRefGoogle Scholar
Morris, D. W., Amenable groups that act on the line, Algebr. Geom. Topol. 6 (2006), 2509–2518; MR 2286034.CrossRefGoogle Scholar
Navas, A., Growth of groups and diffeomorphisms of the interval, Geom. Funct. Anal. 18 (2008), 988–1028; MR 2439001.CrossRefGoogle Scholar
Ornstein, D., Bernoulli shifts with the same entropy are isomorphic, Adv. Math. 4 (1970), 337–352; MR 0257322.CrossRefGoogle Scholar
Ornstein, D., Two Bernoulli shifts with infinite entropy are isomorphic, Adv. Math. 5 (1970), 339–348; MR 0274716.CrossRefGoogle Scholar
Ornstein, D. S. and Weiss, B., Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math. 48 (1987), 1–141; MR 910005.CrossRefGoogle Scholar
Popa, S., Some computations of $1$-cohomology groups and construction of non-orbit-equivalent actions, J. Inst. Math. Jussieu 5 (2006), 309–332; MR 2225044.CrossRefGoogle Scholar
Popa, S. and Sasyk, R., On the cohomology of Bernoulli actions, Ergodic Theory Dynam. Systems 27 (2007), 241–251; MR 2297095.CrossRefGoogle Scholar
Rudolph, D. J. and Schmidt, K., Almost block independence and Bernoullicity of $\textbf {Z}^d$-actions by automorphisms of compact abelian groups, Invent. Math. 120 (1995), 455–488; MR 1334481.CrossRefGoogle Scholar
Rourke, C. and Wiest, B., Order automatic mapping class groups, Pacific J. Math. 194 (2000), 209–227; MR 1756636.CrossRefGoogle Scholar
Schmidt, K., Amenability, Kazhdan's property $T$, strong ergodicity and invariant means for ergodic group-actions, Ergodic Theory Dynam. Systems 1 (1981), 223–236; MR 661821.CrossRefGoogle Scholar
Schmidt, K. and Verbitskiy, E., Abelian sandpiles and the harmonic model, Comm. Math. Phys. 292 (2009), 721–759; MR 2551792.CrossRefGoogle Scholar
Seward, B., Bernoulli shifts with bases of equal entropy are isomorphic, Preprint (2018), arXiv:1805.08279.Google Scholar
Seward, B., Positive entropy actions of countable groups factor onto Bernoulli shifts, J. Amer. Math. Soc. 33 (2020), 57–101; MR 4066472.CrossRefGoogle Scholar
Short, H. and Wiest, B., Orderings of mapping class groups after Thurston, Enseign. Math. (2) 46 (2000), 279–312; MR 1805402.Google Scholar
Stepin, A. M., Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR 223 (1975), 300–302; MR 0409769.Google Scholar
Takesaki, M., Theory of operator algebras. I, Encyclopaedia of Mathematical Sciences, vol. 124, Operator Algebras and Non-commutative Geometry, vol. 5 (Springer, Berlin, 2002). Reprint of the first (1979) edition; MR 1873025.Google Scholar
Takesaki, M., Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Operator Algebras and Non-commutative Geometry, vol. 6 (Springer, Berlin, 2003); MR 1943006.Google Scholar
Tucker-Drob, R., Mixing actions of countable groups are almost free, Proc. Amer. Math. Soc. 143 (2015), 5227–5232; MR 3411140.CrossRefGoogle Scholar
Varopoulos, N. T., Wiener-Hopf theory and nonunimodular groups, J. Funct. Anal. 120 (1994), 467–483; MR 1266317.CrossRefGoogle Scholar
Vinogradov, A. A., On the free product of ordered groups, Mat. Sb. N.S. 25 (1949), 163–168; MR 0031482.Google Scholar