On Magnus' Freiheitssatz and free polynomial algebras

Document Type : Ischia Group Theory 2014

Authors

1 Fairfield University

2 University of Passau

3 University of Hamburg

Abstract

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.

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References

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International Journal of Group Theory
Volume 4, Issue 1 - Serial Number 1
Proceedings of the Ischia Group Theory 2014-Part I
March 2015
Pages 13-19

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