On Bass' ``Strong conjecture'' about projective modules
Let \({\mathbb{Z}}[G]\) be the integral group ring of a group G. Let P be a finitely generated projective \({\mathbb{Z}}[G]\)-module. It has a rank \(r_ p\) which can be viewed as an integer valued function on conjugacy classes of G. The Bass' ``Strong Conjecture'' [\textit{H. Bass}, Invent. Math. 35, 155- 196 (1976; Zbl 0365.20008), p. 156] claims that \(r_ p(g)=0\) for \(g\neq 1\) in G. Bass proved this conjecture for a class of torsion free groups including those with faithful linear representations. In the present paper the author proves the conjecture for a large class of groups which satisfy the following condition called WD: Suppose H is a finitely generated subgroup of G, \(s\in H\), N a positive integer and s is conjugate in H to \(s^{p^ N}\) for all primes p; then \(s=1\). The author shows that the following groups G satisfy condition WD: (i) G is a torsion group. (ii) G is a nilpotent-by-Noetherian group. (iii) (cf. \textit{H. Bass} [loc. cit.]) G is a linear group. (iv) (cf. \textit{P. A. Linnell} [Proc. Lond. Math. Soc., III. Ser. 47, 83-127 (1983; Zbl 0531.20002)]) The group G does not contain any subgroup isomorphic to the additive group of rational numbers. (v) Every finitely generated subgroup of G satisfies condition WD. (vi) G is a finite extension of some WD-groups. (vii) G is a subdirect product of (maybe infinitely many) WD-groups. (viii) G is a free product of (maybe infinitely many) WD-groups.
finitely generated projective \({\mathbb{Z}}[G]\)-module
integer valued function on conjugacy classes
scientific article; zbMATH DE number 4127430
scientific article