# Invariant Schreier decorations of unimodular random networks

## TL;DR

This paper demonstrates that every 2d-regular unimodular random network can be equipped with an invariant Schreier decoration, linking it to invariant random subgroups of free groups, with implications for graphings and group actions.

## Contribution

It establishes the existence of invariant Schreier decorations for unimodular random networks, extending finite graph results to the measurable setting with new coloring theorems.

## Key findings

- Every 2d-regular unimodular random network has an invariant Schreier decoration.
- Every 2d-regular graphing is locally isomorphic to a graphing from an F_d action.
- Developed measurable coloring theorems for graphings.

## Abstract

We prove that every $2d$-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group $F_d$. As a corollary we get that every $2d$-regular graphing is the local isomorphic image of a graphing coming from a p.m.p. action of $F_d$. The key ingredients of the analogous statement for finite graphs do not generalize verbatim to the measurable setting. We find a more subtle way of adapting these ingredients and prove measurable coloring theorems for graphings along the way.