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On Magnus' Freiheitssatz and free polynomial algebras | ||
| International Journal of Group Theory | ||
| مقاله 4، دوره 4، شماره 1، خرداد 2015، صفحه 13-19 اصل مقاله (188.55 K) | ||
| نوع مقاله: Ischia Group Theory 2014 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2015.7279 | ||
| نویسندگان | ||
| Benjamin Fine* 1؛ Martin Kreuzer2؛ Gerhard Rosenberger3 | ||
| 1Fairfield University | ||
| 2University of Passau | ||
| 3University of Hamburg | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Freiheitssatz؛ one-relator group؛ Magnus representation؛ formal power series rings | ||
| مراجع | ||
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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
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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
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