Context-free pairs of groups II — Cuts, tree sets, and random walks

This is a continuation of the study, begun by Ceccherini-Silberstein and Woess (2009) [5], of context-free pairs of groups and the related context-free graphs in the sense of Muller and Schupp (1985) [22]. The graphs under consideration are Schreier graphs of a subgroup of some finitely generated group, and context-freeness relates to a tree-like structure of those graphs. Instead of the cones of Muller and Schupp (1985) [22] (connected components resulting from deletion of finite balls with respect to the graph metric), a more general approach to context-free graphs is proposed via tree sets consisting of cuts of the graph, and associated structure trees. The existence of tree sets with certain “good” properties is studied. With a tree set, a natural context-free grammar is associated. These investigations of the structure of context free pairs, resp. graphs are then applied to study random walk asymptotics via complex analysis. In particular, a complete proof of the local limit theorem for return probabilities on any virtually free group is given, as well as on Schreier graphs of a finitely generated subgoup of a free group. This extends, respectively completes, the significant work of Lalley (1993, 2001) [18,20].