Cutting up graphs revisited - a short proof of Stallings' structure theorem

TL;DR

This paper presents a concise proof of Stallings' theorem on the structure of finitely generated groups with multiple ends, using automorphism-invariant tree decompositions based on edge removal.

Contribution

It provides a new, short combinatorial proof of the main theorem in classical structure tree theory, extending Dunwoody's and Stallings' results.

Findings

Existence of automorphism-invariant tree decompositions

Extension of Dunwoody's and Stallings' theorems

Simplified combinatorial proof of group structure with multiple ends

Abstract

This is a new and short proof of the main theorem of classical structure tree theory. Namely, we show the existence of certain automorphism-invariant tree-decompositions of graphs based on the principle of removing finitely many edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem on the structure of finitely generated groups with more than one end.