Structure trees, networks and almost invariant sets

TL;DR

This paper develops the theory of structure trees for edge cuts in networks, extending classical theorems like Max-Flow Min-Cut and Stallings' Theorem to infinite networks and group structures, with applications to Sageev cubings.

Contribution

It introduces a comprehensive framework for structure trees in networks, generalizes key theorems to infinite cases, and provides new insights into group structures and cubings.

Findings

Generalized Max-Flow Min-Cut to infinite networks

Proved a conjecture of Kropholler

Provided a relative version of Stallings' Theorem

Abstract

A self-contained account of the theory of structure trees for edge cuts in networks is given. Applications include a generalisation of the Max-Flow Min-Cut Theorem to infinite networks and a short proof of a conjecture of Kropholler. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, which provides information about the structure of the Sageev cubing.