# Relation modules of infinite groups, II

Open Mathematics (2014)

- Volume: 12, Issue: 3, page 436-444
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topMartin Evans. "Relation modules of infinite groups, II." Open Mathematics 12.3 (2014): 436-444. <http://eudml.org/doc/269689>.

@article{MartinEvans2014,

abstract = {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.},

author = {Martin Evans},

journal = {Open Mathematics},

keywords = {Group presentations; Relation module; Stable isomorphism; Non-cancellation; Hurwitz-Radon system; finitely generated groups; presentations; relation modules; exact sequences; augmentation ideals; stable isomorphisms; non-cancellation; Hurwitz-Radon systems},

language = {eng},

number = {3},

pages = {436-444},

title = {Relation modules of infinite groups, II},

url = {http://eudml.org/doc/269689},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Martin Evans

TI - Relation modules of infinite groups, II

JO - Open Mathematics

PY - 2014

VL - 12

IS - 3

SP - 436

EP - 444

AB - 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.

LA - eng

KW - Group presentations; Relation module; Stable isomorphism; Non-cancellation; Hurwitz-Radon system; finitely generated groups; presentations; relation modules; exact sequences; augmentation ideals; stable isomorphisms; non-cancellation; Hurwitz-Radon systems

UR - http://eudml.org/doc/269689

ER -

## References

top- [1] Adams J.F., Vector fields on spheres, Ann. of Math., 1962, 75, 603–632 http://dx.doi.org/10.2307/1970213 Zbl0112.38102
- [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 Zbl0132.01205
- [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 Zbl0857.20010
- [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 Zbl0931.20028
- [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 Zbl1283.20026
- [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 Zbl0279.13007
- [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 Zbl0022.31403
- [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 Zbl0326.20028
- [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 Zbl0247.20026
- [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 Zbl0109.41601
- [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