Freiheitssatz

In mathematics, the Freiheitssatz (German: "freedom/independence theorem": Freiheit + Satz) is a result in the presentation theory of groups, stating that certain subgroups of a one-relator group are free groups.

Statement
Consider a group presentation
 * $$G = \langle x_{1}, \dots, x_{n} | r = 1 \rangle$$

given by $n$ generators $x_{i}$ and a single cyclically reduced relator $r$. If $x_{1}$ appears in $r$, then (according to the freiheitssatz) the subgroup of $G$ generated by $x_{2}, ..., x_{n}$ is a free group, freely generated by $x_{2}, ..., x_{n}$. In other words, the only relations involving $x_{2}, ..., x_{n}$ are the trivial ones.

History
The result was proposed by the German mathematician Max Dehn and proved by his student, Wilhelm Magnus, in his doctoral thesis. Although Dehn expected Magnus to find a topological proof, Magnus instead found a proof based on mathematical induction and amalgamated products of groups. Different induction-based proofs were given later by and.

Significance
The freiheitssatz has become "the cornerstone of one-relator group theory", and motivated the development of the theory of amalgamated products. It also provides an analogue, in non-commutative group theory, of certain results on vector spaces and other commutative groups.