A Magnus theorem for some one-relator groups

TL;DR

This paper proves that certain one-relator groups, including fundamental groups of nonorientable surfaces of genus greater than 3, have the Magnus property, extending previous results for orientable surfaces.

Contribution

It establishes the Magnus property for some nonorientable surface groups, broadening the class of groups known to possess this property.

Findings

Some nonorientable surface groups have the Magnus property.

The result extends previous work on orientable surfaces.

The paper includes a proof for groups with specific relator structures.

Abstract

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed nonorientable surfaces of genus g>3 possess this property. The analogous result for orientable surfaces of any finite genus was obtained by the first author [Geometric methods in group theory, Contemp. Math, 372 (2005) 59-69].