# Freiheitssatz

*independence theorem*

A theorem originally proposed by M. Dehn in a geometrical setting and originally proven by W. Magnus [a1]. This theorem is the cornerstone of one-relator group theory.

The Freiheitssatz says the following: Let be a group defined by a single cyclically reduced relator . If appears in , then the subgroup of generated by is a free group, freely generated by .

In coarser language, the theorem says that if is as above, then the only relations in are the trivial ones.

The Freiheitssatz can be considered as a non-commutative analogue of certain more transparent results in commutative algebra. For example, suppose that is a linear space over a field with a basis . If is the subspace of generated by a vector with , then the elements are linearly independent modulo .

Magnus' method of proof of the Freiheitssatz relies on amalgamations (cf. also Amalgam; Amalgam of groups). This method initiated the use of these products in the study of infinite discrete groups.

One of the by-products of Magnus' proof was an extraordinary description of the structure of these groups, which allowed him to deduce that one-relator groups have solvable word problem (cf. also Identity problem; [a2]).

There are two general approaches to extending the Freiheitssatz. The first is concerned with the notion of the one-relator product of a family of groups, where the element is cyclically reduced and of syllable length at least and is its normal closure in . Some authors (see [a3]) give conditions for the factors to inject into .

The second approach is concerned with multi-relator versions of the Freiheitssatz (see [a3] for a list of references). For example, the following strong result by N.S. Romanovskii [a4] holds: Let have deficiency . Then there exist a subset of of the given generators which freely generates a subgroup of .

#### References

[a1] | W. Magnus, "Über discontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz)" J. Reine Angew. Math. , 163 (1930) pp. 141–165 |

[a2] | W. Magnus, "Das Identitätsproblem für Gruppen mit einer definierenden Relation" Math. Ann. , 106 (1932) pp. 295–307 |

[a3] | B. Fine, G. Rosenberger, "The Freiheitssatz and its extensions" Contemp. Math. , 169 (1994) pp. 213–252 |

[a4] | N.S. Romanovskii, "Free subgroups of finitely presented groups" Algebra i Logika , 16 (1977) pp. 88–97 (In Russian) |

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