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2014 | 12 | 3 | 436-444
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Relation modules of infinite groups, II

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EN
Let F n denote the free group of rank n and d(G) the minimal number of generators of the finitely generated group G. Suppose that R ↪ F m ↠ G and S ↪ F m ↠ G are presentations of G and let $$\bar R$$ and $$\bar S$$ denote the associated relation modules of G. It is well known that $$\bar R \oplus (\mathbb{Z}G)^{d(G)} \cong \bar S \oplus (\mathbb{Z}G)^{d(G)}$$ even though it is quite possible that . However, to the best of the author’s knowledge no examples have appeared in the literature with the property that . Our purpose here is to exhibit, for each integer k ≥ 1, a group G that has presentations as above such that . Our approach depends on the existence of nonfree stably free modules over certain commutative rings and, in particular, on the existence of certain Hurwitz-Radon systems of matrices with integer entries discovered by Geramita and Pullman. This approach was motivated by results of Adams concerning the number of orthonormal (continuous) vector fields on spheres.
Wydawca
Czasopismo
Rocznik
Tom
12
Numer
3
Strony
436-444
Opis fizyczny
Daty
wydano
2014-03-01
online
2013-12-21
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autor
Bibliografia
  • [1] Adams J.F., Vector fields on spheres, Ann. of Math., 1962, 75, 603–632 http://dx.doi.org/10.2307/1970213
  • [2] Auslander L., Schenkman E., Free groups, Hirsch-Plotkin radicals, and applications to geometry, Proc. Amer. Math. Soc., 1965, 16(4), 784–788 http://dx.doi.org/10.1090/S0002-9939-1965-0180596-7
  • [3] Evans M.J., Presentations of groups involving more generators than are necessary, Proc. London Math. Soc., 1993, 67(1), 106–126 http://dx.doi.org/10.1112/plms/s3-67.1.106
  • [4] Evans M.J., Relation modules of infinite groups, Bull. London Math. Soc., 1999, 31(2), 154–162 http://dx.doi.org/10.1112/S0024609398005165
  • [5] Evans M.J., Presentations of groups involving more generators than are necessary II, In: Combinatorial Group Theory, Discrete Groups, and Number Theory, Contemp. Math., 421, American Mathematical Society, Providence, 2006, 101–112 http://dx.doi.org/10.1090/conm/421/08029
  • [6] Evans M.J., Nielsen equivalence classes and stability graphs of finitely generated groups, In: Ischia Group Theory 2006, Ischia, March 29–April 1, 2006, World Scientific, Hackensack, 2007, 103–119
  • [7] Evans M.J., Nielsen equivalence classes of free abelianized extensions of groups, Israel J. Math., 2012, 191(1), 185–207 http://dx.doi.org/10.1007/s11856-011-0211-5
  • [8] Geramita A.V., Pullman N.J., A theorem of Hurwitz and Radon and orthogonal projective modules, Proc. Amer. Math. Soc., 1974, 42(1), 51–56 http://dx.doi.org/10.1090/S0002-9939-1974-0332764-4
  • [9] Gruenberg K.W., Relation Modules of Finite Groups, CBMS Regional Conf. Ser. in Math., 25, American Mathematical Society, Providence, 1976
  • [10] Magnus W., On a theorem of Marshall Hall, Ann. of Math., 1939, 40(4), 764–768 http://dx.doi.org/10.2307/1968892
  • [11] Passi I.B.S., Annihilators of relation modules ¶ II, J. Pure Appl. Algebra, 1975, 6(3), 235–237 http://dx.doi.org/10.1016/0022-4049(75)90018-3
  • [12] Remeslennikov V.N., Sokolov V.G., Some properties of a Magnus embedding, Algebra Logic, 1970, 9(5), 342–349 http://dx.doi.org/10.1007/BF02321898
  • [13] Robinson D.J.S., A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math., 80, Springer, New York, 1996 http://dx.doi.org/10.1007/978-1-4419-8594-1
  • [14] Swan R.G., Vector bundles and projective modules, Trans. Amer. Math. Soc., 1962, 105(2), 264–277 http://dx.doi.org/10.1090/S0002-9947-1962-0143225-6
  • [15] Williams J.S., Free presentations and relation modules of finite groups, J. Pure Appl. Algebra, 1973, 3(3), 203–217 http://dx.doi.org/10.1016/0022-4049(73)90010-8
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-013-0355-0
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