Freiheitssatz for amalgamated products of free groups over maximal cyclic subgroups

TL;DR

This paper extends the classical Freiheitssatz to amalgamated products of free groups over maximal cyclic subgroups, showing conditions under which factors embed into certain quotients.

Contribution

It introduces a Freiheitssatz for amalgamated free products of free groups over maximal cyclic subgroups, generalizing Magnus's original result.

Findings

Factors embed into quotients if element is not conjugate to factors

Generalization of Freiheitssatz to amalgamated free products

Conditions for embedding in quotient groups

Abstract

In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let $F$ be a free group with basis $\mathcal{X}$ and let $r$ be a cyclically reduced element of $F$ which contains a basis element $x \in \mathcal{X}$, then every non-trivial element of the normal closure of $r$ in $F$ contains the basis element $x$. Equivalently, the subgroup freely generated by $\mathcal{X} \backslash \{x\}$ embeds canonically into the quotient group $F / \langle \! \langle r \rangle \! \rangle_{F}$. In this article, we want to introduce a Freiheitssatz for amalgamated products $G=A \ast_{U} B$ of free groups $A$ and $B$, where $U$ is a maximal cyclic subgroup in $A$ and $B$: If an element $r$ of $G$ is neither conjugate to an element of $A$ nor $B$, then the factors $A$, $B$ embed canonically into $G / \langle \! \langle r \rangle \! \rangle_{G}$.