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Furstenberg entropy of intersectional invariant random subgroups

Part of: Stochastic processes Ergodic theory

Published online by Cambridge University Press:  17 September 2018

Yair Hartman  and
Ariel Yadin
Yair Hartman
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA email [email protected]
Ariel Yadin
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, 8410501 Be’er Sheva, Israel email [email protected]
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Abstract

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply, for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.

Keywords

stationary actionsFurstenberg entropyinvariant random subgroup

MSC classification

Primary: 37A40: Nonsingular (and infinite-measure preserving) transformations 37A15: General groups of measure-preserving transformations 37A50: Relations with probability theory and stochastic processes 60G10: Stationary processes
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 10 , October 2018 , pp. 2239 - 2265
Copyright
© The Authors 2018 

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Footnotes

Y. Hartman is supported by the Fulbright Post-Doctoral Scholar Program. A. Yadin is supported by the Israel Science Foundation (grant no. 1346/15). We would like to thank Lewis Bowen and Yair Glasner for helpful discussions. Thanks to an anonymous referee for comments and suggestions.

References

Abert, M., Bergeron, N., Biringer, I., Gelander, T., Nikolov, N., Raimbault, J. and Samet, I., On the growth of L 2 -invariants for sequences of lattices in Lie groups , Ann. Math. 185 (2017), 711–790.Google Scholar
Abért, M., Glasner, Y. and Virág, B., Kestenś theorem for invariant random subgroups , Duke Math. J. 163 (2014), 465–488.Google Scholar
Abert, M., Glasner, Y. and Virag, B., The measurable Kesten theorem , Ann. Probab. 44 (2016), 1601–1646.Google Scholar
Bowen, L., Random walks on random coset spaces with applications to Furstenberg entropy , Invent. Math. 196 (2014), 485–510.Google Scholar
Bowen, L., Invariant random subgroups of the free group , Groups Geom. Dyn. 9 (2015), 891–916.Google Scholar
Bowen, L., Grigorchuk, R. and Kravchenko, R., Invariant random subgroups of lamplighter groups , Israel J. Math. 207 (2015), 763–782.Google Scholar
Bowen, L., Hartman, Y. and Tamuz, O., Generic stationary measures and actions , Trans. Amer. Math. Soc. 369 (2017), 4889–4929.Google Scholar
Burton, P., Lupini, M. and Tamuz, O., Weak equivalence of stationary actions and the entropy realization problem, Preprint (2016), arXiv:1603.05013.Google Scholar
Choquet, G. and Deny, J., Sur lequation de convolution mu= mu-star-sigma , C. R. Acad. Sci. Paris 250 (1960), 799–801.Google Scholar
Coulhon, T. and Saloff-Coste, L., Isopérimétrie pour les groupes et les variétés , Rev. Mat. Iberoam. 9 (1993), 293–314.Google Scholar
Cover, T. M. and Thomas, J. A., Elements of information theory (John Wiley & Sons, 2012).Google Scholar
Furman, A., Random walks on groups and random transformations, Handbook of Dynamical Systems vol. 1 (Elsevier Science, 2002), 931–1014.Google Scholar
Furstenberg, H., Random walks and discrete subgroups of Lie groups , Adv. Probab. Relat. Top. 1 (1971), 1–63.Google Scholar
Furstenberg, H., Boundary theory and stochastic processes on homogeneous spaces , in Harmonic analysis on homogeneous spaces, Proceedings of Symposia in Pure Mathematics, vol. 26 (American Mathematical Society, Providence, RI, 1973), 193–229.Google Scholar
Furstenberg, H. and Glasner, E., Stationary dynamical systems , in Dynamical numbers—interplay between dynamical systems and number theory, Contemporary Mathematics, vol. 532 (American Mathematical Society, Providence, RI, 2010), 1–28; MR 2762131.Google Scholar
Hartman, Y. and Tamuz, O., Furstenberg entropy realizations for virtually free groups and lamplighter groups , J. Anal. Math. 126 (2015), 227–257.Google Scholar
Hartman, Y. and Tamuz, O., Stabilizer rigidity in irreducible group actions , Israel J. Math. 216 (2016), 679–705.Google Scholar
Kaimanovich, V. A., The poisson boundary of amenable extensions , Monatsh. Math. 136 (2002), 9–15.Google Scholar
Kaimanovich, V. A., Amenability and the liouville property , Israel J. Math. 149 (2005), 45–85.Google Scholar
Kaimanovich, V. A. and Vershik, A. M., Random walks on discrete groups: boundary and entropy , Ann. Probab. 11 (1983), 457–490.Google Scholar
Leemann, P.-H., Schreir graphs: transitivity and coverings , Internat. J. Algebra Comput. 26 (2016), 69–93.Google Scholar
Lyons, R. and Peres, Y., Probability on trees and networks, Vol. 42 (Cambridge University Press, 2016).Google Scholar
Morris, B. and Peres, Y., Evolving sets, mixing and heat kernel bounds , Probab. Theory Related Fields 133 (2005), 245–266.Google Scholar
Nevo, A., The spectral theory of amenable actions and invariants of discrete groups , Geom. Dedicata 100 (2003), 187–218.Google Scholar
Nevo, A. and Zimmer, R. J., Rigidity of Furstenberg entropy for semisimple Lie group actions , Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 321–343.Google Scholar
Pete, G., Probability and geometry on groups, http://math.bme.hu/∼gabor/PGG.pdf.Google Scholar
Raugi, A., A general Choquet–Deny theorem for nilpotent groups , Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004), 677–683.Google Scholar
Vershik, A. M., Totally nonfree actions and the infinite symmetric group , Mosc. Math. J. 12 (2012), 193–212.Google Scholar