László Márton Tóth: Invariant Schreier decorations on unimodular random graphs

Speaker: László Márton Tóth

Title: Invariant Schreier decorations on unimodular random graphs

Abstract: It is a nice exercise in combinatorics to show that all finite 2d-regular graphs are edge-disjoint unions of 2-regular graphs on the same vertex set. Equivalently, 2d-regular graphs are Schreier graphs of the free group on d generators. We will consider the analogous problem for unimodular random graphs, where we try to find a Schreier labeling in an invariant random way.

We show that any 2d-regular unimodular random network can be given an invariant random Schreier decoration. Equivalently, every 2d-regular graphing is a local ismorphic image of a graphing coming from a probability measure preserving action of the free goup. Connections to Borel combinatorics, and invariant random subgroups will also be explored.