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# Higher K -theory of group-rings of virtually infinite cyclic groups

(Impact Factor: 1.13). 03/2003; 325(4):711-726. DOI: 10.1007/s00208-002-0397-2

ABSTRACT

F.T. Farrell and L.E. Jones conjectured in [7] that Algebraic K-theory of virtually cyclic subgroups V should constitute building blocks' for the Algebraic K-theory of an arbitrary group G. In [6], they obtained some results on lower K-theory of V. In this paper, we obtain results on higher K-theory of virtually infinite cyclic groups V in the two cases: (i) when V admits an epimorphism (with finite kernel) to the infinite cyclic group (see 2.1 and 2.2(a),(b)) and (ii) when V admits an epimorphism (with finite kernel) to the infinite dihedral group (see 3.1, 3.2, 3.3).

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• "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. "
##### Article: Higher Algebraic K-Theory of p-Adic Orders and Twisted Polynomial and Laurent Series Rings Over p-Adic Orders
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ABSTRACT: Let F be a p-adic field (i.e., any finite extension of ), R the ring of integers of F, Λ any R-order in a semisimple F-algebra Σ, α: Λ → Λ an R-automorphism of Λ, T = ⟨ t ⟩, the infinite cyclic group, Λα[t], the α-twisted polynomial ring over Λ and Λα[T], the α-twisted Laurent series ring over Λ. In this article, we study higher K-theory 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 p-group if Σ is a direct product of matrix algebra over fields, in partial answer to an open question whether is a finite p-group 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 p-torsion group and also that for n ≥ 2 there exists isomorphisms Finally, we prove that NKn(Λα[T]) is p-torsion. 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.
Communications in Algebra 10/2011; 39(10-10):3801-3812. DOI:10.1080/00927872.2011.572470
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• "For R D ‫ޚ‬ the considered Nil-groups 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]). "
##### Article: Non-finiteness results for Nil-groups
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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 Nil-groups include the Nil-groups appearing in the decomposition of K i of virtually cyclic groups for i Ä 1.
Algebraic & Geometric Topology 12/2007; 7(4). DOI:10.2140/agt.2007.7.1979
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• "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]. "
##### Article: Induction Theorems and Isomorphism Conjectures for K- and L-Theory
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ABSTRACT: The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory 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.
Forum Mathematicum 05/2004; 19(3). DOI:10.1515/FORUM.2007.016

Has anyone proved an isomorphsim between the Nil groups of Bass and those of Waldhausen for all n > 1?
As matters arising from my joint paper with G. Tang, " Higher K-theory of group-rings of virtually infinite cyclic groups" Math Annalen 325 (2003) 711-725, I am curious to know if anyone has since proved that the Nil groups of Bass and those of Waldhausen are isomorphic for all n > 1. Such an isomorphism is well known for n less than equal to 1 and we used it to show the vanishing of lower Waldhausen nil groups for regular rings.
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##### Article: Higher K -theory of group-rings of virtually infinite cyclic groups
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ABSTRACT:  F.T. Farrell and L.E. Jones conjectured in [7] that Algebraic K-theory of virtually cyclic subgroups V should constitute building blocks' for the Algebraic K-theory of an arbitrary group G. In [6], they obtained some results on lower K-theory of V. In this paper, we obtain results on higher K-theory of virtually infinite cyclic groups V in the two cases: (i) when V admits an epimorphism (with finite kernel) to the infinite cyclic group (see 2.1 and 2.2(a),(b)) and (ii) when V admits an epimorphism (with finite kernel) to the infinite dihedral group (see 3.1, 3.2, 3.3).
Mathematische Annalen 03/2003; 325(4):711-726. DOI:10.1007/s00208-002-0397-2

S. Sundar