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.^[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

^ Magnus, Wilhelm (1930). "Über diskontinuierliche Gruppen mit einer definierenden Relation. (Der Freiheitssatz)". J. Reine Angew. Math. 163: 141–165.
^ Stillwell, John (1999). "Max Dehn". In James, I. M. (ed.). History of topology. North-Holland, Amsterdam. pp. 965–978. ISBN 0-444-82375-1. MR 1674906. See in particular p. 973.
^ ^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.
^ ^a ^b V.A. Roman'kov (2001) [1994], "Freiheitssatz", Encyclopedia of Mathematics, EMS Press
^ 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. hdl:2027.42/135658. MR 0294465.
^ Weinbaum, C. M. (1972). "On relators and diagrams for groups with one defining relation". Illinois Journal of Mathematics. 16 (2): 308–322. doi:10.1215/ijm/1256052287. MR 0297849.

[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. (ed.). History of topology. North-Holland, Amsterdam. pp. 965–978. ISBN 0-444-82375-1. MR 1674906. See in particular p. 973.

[ls-3] 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.

[eom-4] V.A. Roman'kov (2001) [1994], "Freiheitssatz", Encyclopedia of Mathematics, EMS Press

[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. hdl:2027.42/135658. MR 0294465.

[6] Weinbaum, C. M. (1972). "On relators and diagrams for groups with one defining relation". Illinois Journal of Mathematics. 16 (2): 308–322. doi:10.1215/ijm/1256052287. MR 0297849.

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