Relation modules of infinite groups, II

Martin Evans

Open Mathematics (2014)

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

Abstract

top
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 R ¯ and S ¯ denote the associated relation modules of G. It is well known that R ¯ ⊕ ( ℤ G ) d ( G ) ≅ S ¯ ⊕ ( 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.

How to cite

top

Martin 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. [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. [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. [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. [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. [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. [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. [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. [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. [9] Gruenberg K.W., Relation Modules of Finite Groups, CBMS Regional Conf. Ser. in Math., 25, American Mathematical Society, Providence, 1976 
  10. [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. [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. [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. [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. [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. [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 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.