On Magnus' Freiheitssatz and free polynomial algebras

Document Type : Ischia Group Theory 2014


1 Fairfield University

2 University of Passau

3 University of Hamburg


The Freiheitssatz of Magnus for one-relator groups is one of the cornerstones of combinatorial group theory. In this short note which is mostly expository we discuss the relationship between the Freiheitssatz and corre-
sponding results in free power series rings over fields. These are related to results of Schneerson not readily available in English. This relationship uses a faithful representation of free groups due to Magnus. Using this method in free polynomial algebras provides a proof of the Freiheitssatz for one-relation monoids. We show how the classical Freiheitssatz depends on a condition on certain ideals in power series rings in noncommuting variables over fields. A proof of this result over fields would provide a completely dif erent proof of the classical Freiheitssatz.


Main Subjects

C‎. ‎Squier and C‎. ‎Wrathall (1983). ‎The Freiheitssatz for One-Relation Monoids. Proc‎. ‎Amer‎. ‎Math‎. ‎Soc.. 89, 423-424 V‎. ‎Dotsenko‎, ‎N‎. ‎Iyudu and D‎. ‎Korytin (2005). ‎An Analogue of the Magnus Problem for Associative Algebras. J‎. ‎Math‎. ‎Sci‎. ‎(N‎. ‎Y.). 131, 6023-6025 B‎. ‎Fine and G‎. ‎Rosenberger (1994). ‎The Freiheitssatz and its Extensions‎, ‎Contemp‎. ‎Math.. Amer‎. ‎Math‎. ‎Soc.. 169, 213-252 P‎. ‎Freyd (1968). ‎Redei's Finiteness Theorem for Commutative Semigroups. Proc‎. ‎Amer‎. ‎Math‎. ‎Soc.. 19 R‎. ‎C‎. ‎Lyndon (1972). ‎On the Freiheitssatz. J‎. ‎London Math‎. ‎Soc‎. ‎(2). 5, 95-101 J‎. ‎Howie (1987). ‎How to Generalize One-Relator Group Theory. Ann‎. ‎of Math‎. ‎Stud.. 111, 53-78 W‎. ‎Magnus (1930). ‎Ueber diskontinuerliche Gruppen mit einer definierend Relation. J‎. ‎Reine u‎. ‎Angnew‎. ‎Math.. 163, 141-165 W‎. ‎Magnus (1935). ‎Beziehungen zwischen Gruppen und Idealen in einem speziallen Ring. Math‎. ‎Ann.. 111, 259-280 A‎. ‎A‎. ‎Mikhalev‎, ‎V‎. ‎Shpilrain and J‎. ‎T‎. ‎Yu (2004). Combinatorial Methods‎: ‎Free Groups,Polynomials and Free Algebras. ‎Springer Verlag‎, ‎New York. L‎. ‎M‎. ‎Shneerson (1974). ‎On Free Subsemigroups of finitely presented semigroups. Siberian Math‎. ‎J.. 5, 450-454 L‎. ‎M‎. ‎Shneerson (1979). ‎Conditions Under Which Finitely Presented Algebras are free. Izv‎. ‎Vyssh‎. ‎Uchebn‎. ‎Zaved‎. ‎Mat.. (7), 66-68 L‎. ‎M‎. ‎Shneerson (1980). ‎Condition for Freedom for Finitely Presented Algebras. preprint VINITI 3694-80. , 1-8 L‎. ‎M‎. ‎Shneerson (1988). ‎Conditions Under Which Finitely Presented Algebras are free‎. ‎II.‎, ‎Algebraic and Discrete Systems. Ivanov‎. ‎Gos‎. ‎Univ.‎, ‎Ivanovo. 137, 130-132 L‎. ‎M‎. ‎Shneerson (1972). ‎Identities in one-relation Semigroups. Uchen‎. ‎Zap‎. ‎Ivanov‎, ‎Gos‎. ‎Pedag‎. ‎Inst.. 1, 139-156