Article
Higher K theory of grouprings of virtually infinite cyclic groups
Abdus Salam International Centre for Theoretical Physics, Trst, Friuli Venezia Giulia, Italy
Mathematische Annalen (Impact Factor: 1.13). 03/2003; 325(4):711726. DOI: 10.1007/s0020800203972 Get notified about updates to this publication Follow publication 
Fulltext
Aderemi Kuku Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

 "Now for each F i there exists (by [12, Theorem 7.5.3 (ii)] or [14]) a natural long exact sequence · · · → G n F i → G n F i → G n F i TT → G n−1 F i → G n−1 F i → (IV) where each G n F i K n F i is a finite Abelian group for n ≥ 2 by [12, Theorem 7.1.12]. So, from IVV above, G n F i TTT is finite for all n ≥ 2, i.e., K n F i TTT G n F i TTT is a finite Abelian group. "
[Show abstract] [Hide abstract] ABSTRACT: Let F be a padic field (i.e., any finite extension of ), R the ring of integers of F, Λ any Rorder in a semisimple Falgebra Σ, α: Λ → Λ an Rautomorphism of Λ, T = ⟨ t ⟩, the infinite cyclic group, Λα[t], the αtwisted polynomial ring over Λ and Λα[T], the αtwisted Laurent series ring over Λ. In this article, we study higher Ktheory of Λ, Λα[t], and Λα[T].More precisely, we prove in Section 1 that for all n ≥ 1, SK2n−1(Λ) is a finite pgroup if Σ is a direct product of matrix algebra over fields, in partial answer to an open question whether is a finite pgroup if G is any finite group. So, the answer is affirmative if splits.We also prove that NKn(Λ; α): = ker(Kn(Λα[t]) → Kn(Λ)) is a ptorsion group and also that for n ≥ 2 there exists isomorphisms Finally, we prove that NKn(Λα[T]) is ptorsion. Note that if G is a finite group and Λ =RG such that α(G) = G, then Λα[T] is the group ring RV where V is a virtually infinite cyclic group of the form V = G ⋊αT, where α is an automorphism of G and the action of the infinite cyclic group T = ⟨ t ⟩ on G is given by α(g) = tgt for all g ∈ G. 
 "For R D ޚ the considered Nilgroups are known to vanish for i Ä 2 (see Farrell and Jones [5]) and are known to be n–torsion for an arbitrary group of finite order n (see Kuku and Tang [8]). "
[Show abstract] [Hide abstract] ABSTRACT: Generalizing an idea of Farrell we prove that for a ring ƒ and a ring automorphism ˛ of finite order the groups Nil 0 .ƒI ˛/ and all of its p –primary subgroups are either trivial or not finitely generated as an abelian group. We also prove that i and are ring automorphisms such tha ı is of finite order then Nil 0 .ƒI ; ƒ / and all of its p –primary subgroups are either trivial or not finitely generated as an abelian group. These Nilgroups include the Nilgroups appearing in the decomposition of K i of virtually cyclic groups for i Ä 1. 
 "In this case one gets more precise information as discussed in detail in [22, Section 8]. The results presented there are based on [1], [13], [17] and [19]. The next result is due to Mislin and Matthey [23] for the complex case. "
[Show abstract] [Hide abstract] ABSTRACT: The FarrellJones and the BaumConnes Conjecture say that one can compute the algebraic K and Ltheory of the group ring and the topological Ktheory of the reduced group C^*algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with infinite cyclic kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.