On Cayley graphs of virtually free groups

TL;DR

This paper explores the conditions under which groups are virtually free by analyzing their Cayley graphs, providing new proofs for existing theorems related to finitely presentable and context-free groups.

Contribution

It offers new proofs connecting Cayley graph properties to virtual freeness, expanding understanding of group structures in geometric group theory.

Findings

Cayley graph conditions characterize virtually free groups

New proofs of Dunwoody's theorems on group accessibility

Connections between asymptotic dimension and virtual freeness

Abstract

In 1985, Dunwoody showed that finitely presentable groups are accessible. Dunwoody's result was used to show that context-free groups, groups quasi-isometric to trees or finitely presentable groups of asymptotic dimension 1 are virtually free. Using another theorem of Dunwoody of 1979, we study when a group is virtually free in terms of its Cayley graph and we obtain new proofs of the mentioned results and other previously depending on them.