On the linearity of HNN-extensions with abelian base group

TL;DR

This paper establishes a precise criterion linking Z-linearity and residual finiteness for HNN-extensions with finitely generated abelian base groups, contributing to the understanding of their algebraic structure.

Contribution

It proves that such HNN-extensions are Z-linear exactly when they are residually finite, providing a clear characterization.

Findings

HNN-extensions with finitely generated abelian base groups are Z-linear iff residually finite.

The paper offers a complete characterization of linearity for this class of groups.

It advances the theory connecting residual finiteness and linearity in group extensions.

Abstract

We show that an HNN-extension with finitely generated abelian base group is Z-linear if and only if it is residually finite.