Context-free pairs of groups. II - cuts, tree sets, and random walks

TL;DR

This paper extends the study of context-free pairs of groups by introducing tree sets of cuts and structure trees, providing a new framework to analyze graph structures and their random walk behaviors.

Contribution

It proposes a novel approach using tree sets and structure trees to analyze context-free graphs, advancing understanding of their structure and random walk asymptotics.

Findings

Existence of tree sets with desirable properties established

A natural context-free grammar associated with tree sets

Application to analyze random walk asymptotics using complex analysis

Abstract

This is a continuation of the study, begun by Ceccherini-Silberstein and Woess, of context-free pairs of groups and the related context-free graphs in the sense of Muller and Schupp. Instead of the cones (connected components with respect to 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.