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

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