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

$G=⟨x_1,\dots ,x_n|r=1⟩$

given by Template:Mvar generators x_i and a single cyclically reduced relator Template:Mvar. If x₁ appears in Template:Mvar, then (according to the freiheitssatz) the subgroup of Template:Mvar generated by x₂, ..., x_n is a free group, freely generated by x₂, ..., x_n. In other words, the only relations involving x₂, ..., 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.