Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access August 29, 2016

The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups

  • Tim Austin

Abstract

Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

References

[1] T. Austin. Additivity properties of sofic entropy and measures on model spaces. Preprint, available online at arXiv.org: 1510.02392. Search in Google Scholar

[2] B. Bekka, P. de la Harpe, and A. Valette. Kazhdan’s property (T), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008. 10.1017/CBO9780511542749Search in Google Scholar

[3] L. Bowen. Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc., 23(1):217–245, 2010. 10.1090/S0894-0347-09-00637-7Search in Google Scholar

[4] L. Bowen. Weak isomorphisms between Bernoulli shifts. Israel J. Math., 183:93–102, 2011. 10.1007/s11856-011-0043-3Search in Google Scholar

[5] L. Bowen. Every countably infinite group is almost Ornstein. In Dynamical systems and group actions, volume 567 of Contemp. Math., pages 67–78. Amer. Math. Soc., Providence, RI, 2012. 10.1090/conm/567/11234Search in Google Scholar

[6] L. Bowen. Sofic entropy and amenable groups. Ergodic Theory Dynam. Systems, 32(2):427–466, 2012. 10.1017/S0143385711000253Search in Google Scholar

[7] E. Glasner and B. Weiss. Kazhdan’s property T and the geometry of the collection of invariant measures. Geometric and Functional Analysis, 7:917–935, 1997. 10.1007/s000390050030Search in Google Scholar

[8] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces. Birkhäuser, Boston, 2nd edition, 2001. Search in Google Scholar

[9] Y. Jiang. A remark on T-valued cohomology groups of algebraic group actions. Preprint, available online at arXiv.org: 1509.08278. Search in Google Scholar

[10] T. Kaufman, D. Kazhdan, and A. Lubotzky. Ramanujan complexes and bounded degree topological expanders. In 55th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2014, pages 484–493. IEEE Computer Soc., Los Alamitos, CA, 2014. 10.1109/FOCS.2014.58Search in Google Scholar

[11] D. Kerr. Bernoulli actions of sofic groups have completely positive entropy. Preprint, available online at http://www.math. tamu.edu/~kerr/bernoullicpe20.pdf. Search in Google Scholar

[12] D. Kerr and H. Li. Bernoulli actions and infinite entropy. Groups Geom. Dyn., 5(3):663–672, 2011. 10.4171/GGD/142Search in Google Scholar

[13] D. Kerr and H. Li. Entropy and the variational principle for actions of sofic groups. Invent. Math., 186(3):501–558, 2011. 10.1007/s00222-011-0324-9Search in Google Scholar

[14] D. Kerr and H. Li. Soficity, amenability, and dynamical entropy. Amer. J. Math., 135(3):721–761, 2013. 10.1353/ajm.2013.0024Search in Google Scholar

[15] S. Lang. Topics in cohomology of groups, volume 1625 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1996. Translated from the 1967 French original by the author, Chapter X based on letters written by John Tate. 10.1007/BFb0092624Search in Google Scholar

[16] M. Ledoux. The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001. Search in Google Scholar

[17] D. S. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups. J. AnalyseMath., 48:1–141, 1987. 10.1007/BF02790325Search in Google Scholar

[18] S. Popa. Somecomputations of 1-cohomology groups and construction of non-orbit-equivalent actions. J. Inst.Math. Jussieu, 5(2):309–332, 2006. 10.1017/S1474748006000016Search in Google Scholar

[19] S. Popa and R. Sasyk. On the cohomology of Bernoulli actions. Ergodic Theory Dynam. Systems, 27(1):241–251, 2007 10.1017/S0143385706000502Search in Google Scholar

[20] K. Schmidt. Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions. Ergodic Theory Dynamical Systems, 1(2):223–236, 1981. 10.1017/S014338570000924XSearch in Google Scholar

[21] B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Preprint, available online at arXiv.org: 1405.3604. Search in Google Scholar

[22] B. Seward. Krieger’s finite generator theorem for actions of countable groups II. Preprint, available online at arXiv.org: 1501.03367. Search in Google Scholar

[23] R. J. Zimmer. On the cohomology of ergodic actions of semisimple Lie groups and discrete subgroups. Amer. J. Math., 103(5):937–951, 1981. 10.2307/2374253Search in Google Scholar

Received: 2016-2-19
Accepted: 2016-6-6
Published Online: 2016-8-29

© 2016 Tim Austin

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 23.2.2024 from https://www.degruyter.com/document/doi/10.1515/agms-2016-0006/html
Scroll to top button