On Magnusʹ Freiheitssatz and free polynomial algebras

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The Freiheitssatz of Magnus for one-relator groups is one of thecornerstones of combinatorial group theory. In this short note which is mostlyexpository we discuss the relationship between the Freiheitssatz and corre-sponding results in free power series rings over fields. These are related toresults of Schneerson not readily available in English. This relationship usesa faithful representation of free groups due to Magnus. Using this method infree polynomial algebras provides a proof of the Freiheitssatz for one-relationmonoids. We show how the classical Freiheitssatz depends on a condition oncertain ideals in power series rings in noncommuting variables over fields. Aproof of this result over fields would provide a completely diferent proof of theclassical Freiheitssatz.