Invariant Schreier decorations of unimodular random networks

Tóth, László Márton

Annales Henri Lebesgue, Volume 4 (2021), pp. 1705-1726.

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.

Nous démontrons que tout graphe aléatoire unimodulaire $2d$-régulier admet une décoration de Schreier aléatoire invariante. De manière équivalent, c’est le graphe de Schreier des classes d’un sous-groupe invariant aléatoire du groupe libre $F_d$. Comme corollaire, nous obtenons que tout graphage $2d$-régulier est localement l’image isomorphe d’un graphage provenant d’une action de $F_d$ préservant une mesure de probabilité.

Les ingrédients-clé de l’énoncé analogue pour les graphes finis ne se généralisent pas tels quels au contexte mesurable. Nous trouvons une manière plus subtile d’adapter ces ingrédients, et nous obtenons en chemin des théorèmes de coloriages mesurables de graphages.

Classification: 37A50, 20E05, 05C15
Mots-clés : Schreier graph, graph limits, unimodular random graph, Invariant Random Subgroup

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