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 and denote the associated relation modules of G. It is well known that 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.
@article{bwmeta1.element.doi-10_2478_s11533-013-0355-0,
author = {Martin Evans},
title = {Relation modules of infinite groups, II},
journal = {Open Mathematics},
volume = {12},
year = {2014},
pages = {436-444},
zbl = {1302.20037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0355-0}
}
Martin Evans. Relation modules of infinite groups, II. Open Mathematics, Tome 12 (2014) pp. 436-444. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0355-0/
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