Invariant Schreier decorations of unimodular random networks
Annales Henri Lebesgue, Volume 4 (2021), pp. 1705-1726.

Metadata

KeywordsSchreier graph, graph limits, unimodular random graph, Invariant Random Subgroup

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.


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