A Magnus extension for locally indicable groups

TL;DR

This paper extends the Magnus property, originally proven for surface groups, to a broader class of locally indicable groups, analyzing how direct factors influence this property.

Contribution

It generalizes Magnus extensions from free groups to locally indicable groups and explores the effects of adding direct factors.

Findings

Proves a Magnus extension for locally indicable groups.

Establishes Freiheitssatz versions for these groups.

Analyzes the impact of direct factors on the Magnus property.

Abstract

A group $G$ possesses the Magnus property if for every two elements $u$, $v \in G$ with the same normal closure, $u$ is conjugate to $v$ or $v^{-1}$. O. Bogopolski and J. Howie proved independently that the fundamental groups of all closed orientable surfaces possess the Magnus property. The analogous result for closed non-orientable surfaces was proved by O. Bogopolski and K. Sviridov except for one case that was later covered by the author. In this article, we generalize those results, which can be viewed as Magnus extensions for free groups, to a Magnus extension for locally indicable groups and consider the influence of adding a group as a direct factor. For this purpose, we also prove versions of the Freiheitssatz for locally indicable groups and of a result by M. Edjvet adding a group as a direct factor.