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

• ##### Guoping Tang
Mathematische Annalen (Impact Factor: 1.38). 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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##### 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):3801-3812. · 0.36 Impact Factor
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##### Chapter: In: Group Theory HIGHER ALGEBRAIC K – THEORY OF G – REPRESENTATIONS FOR THE ACTIONS OF FINITE AND ALGEBRAIC GROUPS G
02/2010: pages 41 -82; , ISBN: ISBN-10: : 1 - 60876 - 175 - 4
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##### Article: Higher algebraic K-theory for twisted Laurent series rings over orders and semisimple algebras
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ABSTRACT: Let R be the ring of integers in a number field F, Λ any R-order in a semisimple F-algebra Σ, α an R-automorphism of Λ. Denote the extension of α to Σ also by α. Let Λα [T] (resp. Σα [T] be the α-twisted Laurent series ring over Λ (resp. Σ). In this paper we prove that (i) There exist isomorphisms $\mathbb{Q}\otimes K_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes G_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes K_{n}(\Sigma_{\alpha}[T])$ ) for all n ≥ 1. (ii) $G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)\simeq G_n(\Lambda_{\alpha}[T],\hat{Z}_l)$ is an l-complete profinite Abelian group for all n≥2. (iii) ${\rm div} G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)=0$ for all n≥2. (iv) $G_n(\Lambda_{\alpha}[T]) \longrightarrow G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)$ is injective with uniquely l-divisible cokernel (for all n≥2). (v) K –1(Λ), K –1(Λα [T]) are finitely generated Abelian groups.
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