# A Magnus theorem for some amalgamated products

## TL;DR

This paper proves the Magnus property for certain amalgamated products, including the fundamental group of a non-orientable surface of genus 3, extending previous results for higher genus surfaces.

## Contribution

It establishes the Magnus property for specific amalgamated products, notably including genus 3 non-orientable surface groups, answering an open question in the field.

## Key findings

- Magnus property holds for some amalgamated products
- Includes fundamental groups of non-orientable surfaces of genus 3
- Extends results previously known for genus greater than 3

## 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 in $G$ to $v$ or $v^{-1}$. We prove the Magnus property for some amalgamated products including the fundamental group of a closed non-orientable surface of genus 3. This answers a question of O. Bogopolski and K. Sviridov, who obtained the analogous result for genus $g > 3$.