# 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

${\displaystyle G=\langle x_{1},\dots ,x_{n}|r=1\rangle }$

given by n generators xi and a single cyclically reduced relator r. If x1 appears in r, then (according to the freiheitssatz) the subgroup of G generated by x2, ..., xn is a free group, freely generated by x2, ..., xn. In other words, the only relations involving x2, ..., xn 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.[1] Although Dehn expected Magnus to find a topological proof,[2] Magnus instead found a proof based on mathematical induction[3] and amalgamated products of groups.[4] Different induction-based proofs were given later by Lyndon (1972) and Weinbaum (1972).[3][5][6]

## 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.[4]

## References

1. ^ Magnus, Wilhelm (1930). "Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz)". J. Reine Angew. Math. 163: 141–165.
2. ^ Stillwell, John (1999). "Max Dehn". In James, I. M. History of topology. North-Holland, Amsterdam. pp. 965–978. ISBN 0-444-82375-1. MR 1674906. See in particular p. 973.
3. ^ a b Lyndon, Roger C.; Schupp, Paul E. (2001). Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin. p. 152. ISBN 3-540-41158-5. MR 1812024.
4. ^ a b V.A. Roman'kov (2001), "Freiheitssatz", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
5. ^ Lyndon, Roger C. (1972). "On the Freiheitssatz". Journal of the London Mathematical Society. Second Series. 5: 95–101. doi:10.1112/jlms/s2-5.1.95. MR 0294465.
6. ^ Weinbaum, C. M. (1972). "On relators and diagrams for groups with one defining relation". Illinois Journal of Mathematics. 16: 308–322. MR 0297849.