Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T13:07:18.914Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant random subgroups of semidirect products

Published online by Cambridge University Press:  04 September 2018

IAN BIRINGER ,
LEWIS BOWEN  and
OMER TAMUZ
IAN BIRINGER
Affiliation:
Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467, USA email [email protected]
LEWIS BOWEN
Affiliation:
Mathematics Department, University of Texas at Austin, 1 University Station C1200, 78712, USA
OMER TAMUZ
Affiliation:
Department of Mathematics, California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We study invariant random subgroups (IRSs) of semidirect products $G=A\rtimes \unicode[STIX]{x1D6E4}$. In particular, we characterize all IRSs of parabolic subgroups of $\text{SL}_{d}(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$ are either of the form $\mathbb{R}^{d}\rtimes K$ for some IRS of $\text{SL}_{d}(\mathbb{R})$, or are induced from IRSs of $\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$, where $\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$ is a lattice.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 2 , February 2020 , pp. 353 - 366
Copyright
© Cambridge University Press, 2018 

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.)

References

Abért, 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. of Math. (2) 185(3) (2017), 711–790.Google Scholar
Abért, M., Glasner, Y. and Virág, B.. Kesten’s theorem for invariant random subgroups. Duke Math. J. 163(3) (2014), 465–488.Google Scholar
Bader, U., Duchesne, B., Lécureux, J. and Wesolek, P.. Amenable invariant random subgroups. Israel J. Math. 1 (2014), 1–24.Google Scholar
Bekka, B.. Operator-algebraic superridigity for sl n (z), n ≥ 3. Invent. Math. 169(2) (2007), 401–425.Google Scholar
Benedetti, R. and Petronio, C.. Lectures on Hyperbolic Geometry (Universitext) . Springer, Berlin, 1992.Google Scholar
Biringer, I. and Tamuz, O.. Unimodularity of invariant random subgroups. Trans. Amer. Math. Soc. 369(6) (2017), 4043–4061.Google Scholar
Bowen, L.. Random walks on random coset spaces with applications to Furstenberg entropy. Invent. Math. 196(2) (2014), 485–510.Google Scholar
Bowen, L.. Invariant random subgroups of the free group. Groups Geom. Dyn. 9(3) (2015), 891–916.Google Scholar
Bowen, L., Grigorchuk, R. and Kravchenko, R.. Invariant random subgroups of lamplighter groups. Israel J. Math. 207(2) (2015), 763–782.Google Scholar
Chabauty, C.. Limite d’ensembles et géométrie des nombres. Bull. Soc. Math. France 78 (1950), 143–151.Google Scholar
Diatta, A. and Foreman, B.. Lattices in contact Lie groups and 5-dimensional contact solvmanifolds. Kodai Math. J. 38(1) (2015), 228–248.Google Scholar
Furstenberg, H.. A note on Borel’s density theorem. Proc. Amer. Math. Soc. 55(1) (1976), 209–212.Google Scholar
Hartman, Y. and Tamuz, O.. Furstenberg entropy realizations for virtually free groups and lamplighter groups. J. Anal. Math. 126(1) (2015), 227–257.Google Scholar
Knapp, A. W.. Lie Groups Beyond an Introduction. Springer Science & Business Media, Switzerland, 2013, p. 140.Google Scholar
Nevo, A. and Zimmer, R. J.. A generalization of the intermediate factors theorem. J. Anal. Math. 86(1) (2002), 93–104.Google Scholar
Phelps, R. R.. Lectures on Choquet’s Theorem (Lecture Notes in Mathematics, 1757) , 2nd edn. Springer, Berlin, 2001.Google Scholar
Raghunathan, M. S.. Discrete Subgroups of Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, 68) . Springer, New York, 1972.Google Scholar
Stuck, G. and Zimmer, R. J.. Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2) 3 (1994), 723–747.Google Scholar
Thomas, S. and Tucker-Drob, R.. Invariant random subgroups of strictly diagonal limits of finite symmetric groups. Bull. Lond. Math. Soc. 46(5) (2014), 1007–1020.Google Scholar
Tits, J.. Systemes générateurs de groupes de congruence. C. R. Acad. Sci. Paris AB 283(9) (1976), 693–695.Google Scholar
Tucker-Drob, R. D.. Weak equivalence and non-classifiability of measure preserving actions. Ergod. Th. & Dynam. Sys. 35(01) (2015), 293–336.Google Scholar
Vershik, A. M.. Totally nonfree actions and the infinite symmetric group. Mosc. Math. J. 12(1) (2012), 193–212, 216.Google Scholar