From acyclic groups to the Bass conjecture for amenable groups

Berrick, A. J.; Chatterji, Indira; Mislin, Guido

doi:10.1007/s00208-004-0521-6

Cited by 24 publications

(28 citation statements)

References 33 publications

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“…While revising this paper, we have learned that Berrick, Chatterji and Mislin [7] have proved the strong Bass conjecture for amenable groups in the case k = C. In fact they prove a rather stronger version for the Banach space ℓ 1 (G); on the other hand their results do not apply to the case when k has nonzero characteristic. Their techniques are rather different from ours, and depend on recent work of Vincent Lafforgue [26].…”

Section: Conjecture 15 (The Strong Bass Conjecture)mentioning

confidence: 97%

“…commutes with direct limits and we have not used, so far, the assumption that G is a subgroup of GL n (C), the following additional remark is true. To see the "and also" assertion, notice thatK 0 (kN ) is a subgroup of Wh k (N ×Z) [5,Chapter XII,§7] and that N × Z is also both torsion-free and virtually nilpotent.…”

Section: Whitehead Groups Of Solvable Linear Groupsmentioning

confidence: 99%

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Whitehead groups and the Bass conjecture

Farrell

Linnell

2003

Mathematische Annalen

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Section: Conjecture 15 (The Strong Bass Conjecture)mentioning

confidence: 97%

Section: Whitehead Groups Of Solvable Linear Groupsmentioning

confidence: 99%

Whitehead groups and the Bass conjecture

Farrell

Linnell

2003

Mathematische Annalen

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“…Later, this conjecture has been proved for many more groups, notably by Eckmann [13], Emmanouil [15] and Linnell [22]. The latest advances are given in [3] and the first section of the present paper is a quick survey of the Bass conjecture, together with an outline of the proof of the main result of [3].…”

Section: Introductionmentioning

confidence: 94%

“…This in particular implies that for G profinite one has HS(CG⊗ ZG Q)(s) = 0 for all s ∈ G\{1}, because Q cannot be a subgroup of a profinite group. We give an outline of the strategy for proving the main result of [3], which states that the Bost Conjecture implies the Bass Conjecture over C (see Theorem 1.1 below). The Bost Conjecture asserts that the Bost assembly map β G 0 : K G 0 (EG) → K 0 (ℓ 1 G) is an isomorphism (see [20] and [26]).…”

Section: Review Of the Bass Conjecturementioning

confidence: 99%

Hattori-Stallings trace and Euler characteristics for groups

Chatterji¹,

Mislin²

2009

Geometric and Cohomological Methods in Group Theory

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“…Berrick-Chatterji-Mislin [12] prove that a group G satisfies the Bass Conjecture 1.8 for F = C and the Bass Conjecture 1.6 for integral domains for R = Z, if G satisfies the Bost Conjecture. Because the Bost Conjecture is known for many groups, this is also true for the Bass Conjecture for F = C. Since the Bost Conjecture deals with l 2 -spaces, this strategy can only work for subrings of C.…”

Section: Bass Conjecturesmentioning

confidence: 99%

On the Farrell-Jones Conjecture and its applications

Bartels

Lück

Reich

2007

Journal of Topology

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Abstract. We present the status of the Farrell-Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true and study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture and conjectures generalizing Moody's Induction Theorem. Thus we extend the class of groups for which these conjectures are known considerably. Mathematics Subject Classification (2000). 19Dxx, 19A31,19B28Keywords. Algebraic K-theory of group rings with arbitrary coefficients, Farrell-Jones Conjecture, Bass Conjecture, Kaplansky Conjecture, Moody's Induction Theorem. Introduction and statements of results0.1. Background. The Farrell-Jones Conjecture for algebraic K-theory predicts the structure of K n (RG) for a group G and a ring R. There is also an L-theory version. For applications in topology and geometry the case R = Z is the most important one since many topological invariants of manifolds and CW -complexes such as the finiteness obstruction, the Whitehead torsion and the surgery obstruction take values in the algebraic K-or L-theory of the integral group ring Zπ of the fundamental group π. The Farrell-Jones Conjecture for R = Z implies several famous conjectures, e.g., the Novikov Conjecture, (in high dimensions) the Borel Conjecture, and the triviality of compact h-cobordisms with torsionfree fundamental group. On the other hand proofs of the Farrell-Jones Conjecture for certain groups often rely on working with integral coefficients since they are based on these geometric connections. This is the reason why more is known about the algebraic K-and L-theory of ZG than of CG which is in some sense surprising since CG has better ring theoretic properties than ZG. For the status of the Farrell-Jones Conjecture with coefficient in Z we refer for instance to [45, Sections 5.2 and 5.3].Recently the geometric approaches have been generalized so far that they also apply to other coefficient rings than Z (see for instance , BartelsReich [8], Bartels-Lück-Reich [6], Quinn [54]). This is interesting for algebraic and ring theoretic applications, where one would like to consider for example fields, rings of integers in algebraic number fields and integral domains. The purpose of

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From acyclic groups to the Bass conjecture for amenable groups

Abstract: Abstract. We prove that the Bost Conjecture on the ℓ 1 -assembly map for countable discrete groups implies the Bass Conjecture. It follows that all amenable groups satisfy the Bass Conjecture.

Cited by 24 publications

References 33 publications

Whitehead groups and the Bass conjecture

Whitehead groups and the Bass conjecture

Hattori-Stallings trace and Euler characteristics for groups

On the Farrell-Jones Conjecture and its applications

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