Magnus's Freiheitssatz

Remarks

Magnus proved the theorem by induction on the length of $r$. After a change of generators, $G$ is written as an HNN extension or an amalgamated product of one-relator groups with shorter relators, and the induction applies to them. The same method shows that the word problem of one-relator groups is solvable.

Example: in $\Z^2 = \langle a, b \mid a b a^{-1} b^{-1} \rangle$, the theorem says that $a$ generates an infinite cyclic subgroup, a free group of rank $1$. In the Baumslag–Solitar group $\langle a, b \mid b a b^{-1} a^{-2} \rangle$, both $a$ and $b$ have infinite order.

The word $r$ must be cyclically reduced. For $r = b a b^{-1}$, which contains $b$, the group $\langle a, b \mid b a b^{-1} \rangle$ is infinite cyclic generated by $b$, and $a = 1$ in it. So $\set{a}$ omits $b$, but $a$ does not generate a free group of rank $1$. The cyclic reduction of $r$ is $a$, which does not contain $b$.

Going deeper

For a student

The theorem is the starting point of the theory of one-relator groups. Magnus's method of writing a one-relator group as an HNN extension of a group with a shorter relator is the main tool of the area. Suppose that $r$ is cyclically reduced and that every generator in $X$ occurs in $r$. Then every proper subset of $X$ is a free basis of a free subgroup of $G$.