Invariant random subgroups of semidirect products. (English) Zbl 1437.37002
The authors study transverse invariant random subgroups (IRSs) of semidirect products \(G=A \rtimes \Gamma\) when \(A\) is torsion-free abelian or simply connected nilpotent.
In particular, they characterize all IRSs of parabolic subgroups of \(\mathrm{SL}_d(\mathbb {R})\), and show that all ergodic IRSs of \(\mathbb {R}^d \rtimes \mathrm{SL}_d(\mathbb {R})\) are either of the form \(\mathbb {R}^d \rtimes K\) for some IRS of \(\mathrm{SL}_d(\mathbb {R})\), or are induced from IRSs of \(\Lambda \rtimes \mathrm{SL}(\Lambda)\), where \(\Lambda < \mathbb {R}^d\) is a lattice.
As applications they study IRSs of two familiar semidirect products: the special affine groups \(\mathbb {R}^d \rtimes \mathrm{SL}_d(\mathbb {R})\) and the parabolic subgroups of \(\mathrm{SL}_d(\mathbb {R})\).
In particular, they characterize all IRSs of parabolic subgroups of \(\mathrm{SL}_d(\mathbb {R})\), and show that all ergodic IRSs of \(\mathbb {R}^d \rtimes \mathrm{SL}_d(\mathbb {R})\) are either of the form \(\mathbb {R}^d \rtimes K\) for some IRS of \(\mathrm{SL}_d(\mathbb {R})\), or are induced from IRSs of \(\Lambda \rtimes \mathrm{SL}(\Lambda)\), where \(\Lambda < \mathbb {R}^d\) is a lattice.
As applications they study IRSs of two familiar semidirect products: the special affine groups \(\mathbb {R}^d \rtimes \mathrm{SL}_d(\mathbb {R})\) and the parabolic subgroups of \(\mathrm{SL}_d(\mathbb {R})\).
Reviewer: Armand Azonnahin (São Paulo)
MSC:
| 37A15 | General groups of measure-preserving transformations and dynamical systems |
| 37A05 | Dynamical aspects of measure-preserving transformations |
| 37A20 | Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations |
| 22E40 | Discrete subgroups of Lie groups |
| 22D40 | Ergodic theory on groups |
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