Almost Invariant Sets

TL;DR

The paper provides a short proof of Kropholler's conjecture, extends Stallings' Theorem on groups with multiple ends, and generalizes the Almost Stability Theorem to inform the structure of Sageev cubings.

Contribution

It offers a concise proof of a conjecture, a relative version of Stallings' Theorem, and a generalized Almost Stability Theorem for Sageev cubings.

Findings

Proof of Kropholler's conjecture provided

Extended Stallings' Theorem to relative cases

Generalized Almost Stability Theorem for Sageev cubings

Abstract

A short proof of a conjecture of Kropholler is given. 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, that gives information about the structure of the Sageev cubing.