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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 ianbiringer@gmail.com
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
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Abstract

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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.

Information

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

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