2013/09/20 by Jan Cannizzo, Cannizzo, Jan · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS
paper · pdf · doi:10.48550/arxiv.1309.5163
16 pages, added references and figure, to appear in L'Enseignement Mathematique
arxiv created 2014/05/30 · arxiv updated 2014/06/02
Schreier graphs, which possess both a graph structure and a 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 invariant in some sense (e.g. conjugation-invariant, or 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 invariant random subgroups, a field which has recently attracted a great deal of attention.