A descriptive construction of trees and Stallings' theorem

TL;DR

This paper presents a new descriptive construction of trees for multi-ended graphs, providing an alternative proof of Stallings' theorem and extending ideas to countable Borel equivalence relations.

Contribution

It offers a novel descriptive approach to constructing trees in multi-ended graphs and adapts the proof to Borel equivalence relations, leading to new decomposition results.

Findings

Provides a new proof of Stallings' theorem using descriptive methods

Extends tree construction techniques to countable Borel equivalence relations

Establishes conditions for free decomposition and treeability

Abstract

We give a descriptive construction of trees for multi-ended graphs, which yields yet another proof of Stallings' theorem on ends of groups. Even though our proof is, in principle, not very different from already existing proofs and it draws ideas from a paper of Kr\"{o}n, it is written in a way that easily adapts to the setting of countable Borel equivalence relations, leading to a free decomposition result and a sufficient condition for treeability.