Scott and Swarup's regular neighbourhood as a tree of cylinders

TL;DR

This paper reinterprets Scott and Swarup's canonical splitting of a finitely presented group as a tree of cylinders derived from a JSJ splitting, providing a new perspective on the group's structure.

Contribution

The authors offer an alternative construction of Scott and Swarup's regular neighbourhood by demonstrating it as the tree of cylinders of a JSJ splitting.

Findings

Reinterpreted Scott and Swarup's splitting as a tree of cylinders

Connected the regular neighbourhood to JSJ splittings

Provided a new construction method for the canonical splitting

Abstract

Let G be a finitely presented group. Scott and Swarup have constructed a canonical splitting of G which encloses all almost invariant sets over virtually polycyclic subgroups of a given length. We give an alternative construction of this regular neighbourhood, by showing that it is the tree of cylinders of a JSJ splitting.