Context-Free Groups and Their Structure Trees

TL;DR

This paper presents a simplified, elementary construction of a structure tree for groups acting on graphs with finite tree width, leading to a new proof that context-free groups are virtually free, avoiding complex existing theorems.

Contribution

It introduces a direct, elementary construction of the structure tree for groups acting on graphs, simplifying the proof that context-free groups are virtually free.

Findings

Constructed a structure tree directly from the graph's cuts.

Provided a simplified proof of Muller and Schupp's characterization.

Showed a new, elementary approach to accessibility of groups.

Abstract

Let Gamma be a connected, locally finite graph of finite tree width and G be a group acting on it with finitely many orbits and finite node stabilizers. We provide an elementary and direct construction of a tree T on which G acts with finitely many orbits and finite vertex stabilizers. Moreover, the tree is defined directly in terms of the structure tree of optimally nested cuts of Gamma. Once the tree is constructed, standard Bass-Serre theory yields that G is virtually free. This approach simplifies the existing proofs for the fundamental result of Muller and Schupp that characterizes context-free groups as f.g. virtually free groups. Our construction avoids the explicit use of Stallings' structure theorem and it is self-contained. We also give a simplified proof for an important consequence of the structure tree theory by Dicks and Dunwoody which has been stated by Thomassen and Woess. It says that a f.g. group is accessible if and only if its Cayley graph is accessible.