On invariant Schreier structures (Q2349112)

Summary: Schreier graphs, which possess both a graph structure and a \textit{Schreier structure} (an edge-labeling by the generators of a group), are objects of fundamental importance in group theory and geometry. We study the Schreier structures with which unlabeled graphs may be endowed, with emphasis on structures which are \textit{invariant} in some sense (e.g. conjugation-invariant, or \textit{sofic}). We give proofs of a number of ``folklore'' results, such as that every regular graph of even degree admits a Schreier structure, and show that, under mild assumptions, the space of invariant Schreier structures over a given invariant graph structure is very large, in that it contains uncountably many ergodic measures. Our work is directly connected to the theory of \textit{invariant random subgroups}, a field which has recently attracted a great deal of attention.