Certain residual properties of HNN-extensions with central associated subgroups

TL;DR

This paper investigates the residual properties of HNN-extensions with central associated subgroups, providing necessary and sufficient conditions for such extensions to be residually in a broad class of groups.

Contribution

It establishes precise criteria for when HNN-extensions with central associated subgroups are residually within a specified class of groups, expanding understanding of residual properties in group theory.

Findings

Provides necessary and sufficient conditions for residuality

Extends residual property analysis to HNN-extensions with central subgroups

Characterizes residuality in classes closed under various group operations

Abstract

Suppose that $G$ is a group, $H$ and $K$ are proper isomorphic central subgroups of $G$, and $\mathfrak{G}$ is an HNN-extension of $G$ with the associated subgroups $H$ and $K$. We prove necessary and sufficient conditions for $\mathfrak{G}$ to be residually a $\mathcal{C}$-group, where $\mathcal{C}$ is a class of groups closed under taking subgroups, extensions, homomorphic images, and Cartesian products of the form $\prod_{y \in Y}X_{y}$, where $X, Y \in \mathcal{C}$ and $X_{y}$ is an isomorphic copy of $X$ for each $y \in Y$.