# Invariant random subgroups of semidirect products

## TL;DR

This paper characterizes invariant random subgroups of semidirect products, especially in the context of $ ext{SL}_d( ext{R})$ and related groups, revealing their structure and classification.

## Contribution

It provides a complete description of IRSs in certain semidirect products, including parabolic subgroups and the Euclidean group, extending understanding of their subgroup distributions.

## Key findings

- All IRSs of parabolic subgroups of SL_d(R) are characterized.
- Ergodic IRSs of R^d ⋉ SL_d(R) are either of a specific form or induced from lattice subgroups.
- The structure of IRSs in these groups is fully classified.

## Abstract

We study invariant random subgroups (IRSs) of semidirect products $G = A \rtimes \Gamma$. In particular, we 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.