Freiheitssatz for amalgamated products of free groups over maximal cyclic subgroups

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 $F/⟨⟨r⟩{⟩}_F$. 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 $G/⟨⟨r⟩{⟩}_G$.