Adapted almost invariant sets and splittings of groups

TL;DR

This paper extends key concepts of almost invariant sets and group splittings to relative cases, establishing new versions of the algebraic torus theorem, regular neighbourhoods, and JSJ decompositions.

Contribution

It introduces relative versions of fundamental theorems and structures in group theory, including the algebraic torus theorem and JSJ decompositions.

Findings

Proved a relative algebraic torus theorem.

Established existence and uniqueness of relative algebraic regular neighbourhoods.

Extended JSJ decomposition theory to relative settings.

Abstract

We prove relative versions of many earlier results about almost invariant sets and splittings of groups. In particular, we prove a relative version of the algebraic torus theorem, and we prove the existence and uniqueness of relative versions of algebraic regular neighbourhoods and of JSJ decompositions.