Invariant Schreier decorations of unimodular random networks

2019/06/07 by László Márton Tóth, Tóth, László Márton · 1 citation
Computer Science · Mathematics · #05C15 #20E05 #37A50 #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Graph theory and applications #Group Theory (math.GR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1906.03137

openalex publication_date 2019/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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