  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.
