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

Kuku, Aderemi O.; Tang, Guoping

doi:10.1007/s00208-002-0397-2

Cited by 12 publications

(14 citation statements)

References 6 publications

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“…There are two types of virtually infinite cyclic groups -one type of the form V = G α T as described above and the other of the form V = G 0 * H G 1 , where the groups G 0 , G 1 , H are finite and [G 0 : H] = [G 1 : H] = 2. For some results on higher K-theory of both types of groups see [15, 7.5] or [16]. In this paper, we obtain results on higher K-theory of twisted Laurent series ring that translate into results on grouprings RV, V = G α T, as we now explain.…”

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confidence: 55%

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Higher Algebraic K-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

Kuku

2008

Algebr Represent Theor

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confidence: 55%

“…Now, by replacing the maximal order in the proof of [15,Theorem 7.2.3,p. 146] or [16] with the α-invariant order containing , as in Theorem 2, we have that for all n ≥ 1, δ n : …”

Section: Proof Of the Claim Note That A/j A (A /Jmentioning

confidence: 98%

Higher Algebraic K-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

Kuku

2008

Algebr Represent Theor

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“…It is well known that the Nil-groups associated to virtually cyclic groups are either trivial or infinitely generated (see [F77], [G1], [R]). Furthermore, these groups are known to be purely torsion (see [We81], [CP02], [KT03], [G2]), giving us the second statement.…”

Section: Discussionmentioning

confidence: 99%

Splitting formulas for certain Waldhausen Nil-groups

Lafont¹,

Ortiz²

2009

Journal of the London Mathematical Society

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Abstract. We provide splitting formulas for certain Waldhausen Nil-groups. We focus on Waldhausen Nil-groups associated to acylindrical amalgamations Γ = G 1 * H G 2 of groups G 1 , G 2 over a common subgroup H. For these amalgamations, we explain how, provided G 1 , G 2 , Γ satisfy the Farrell-Jones isomorphism conjecture, the Waldhausen Nil-groupscan be expressed as a direct sum of Nil-groups associated to a specific collection of virtually cyclic subgroups of Γ. A special case covered by our theorem is the case of arbitrary amalgamations over a finite group H.

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“…Note that this article is the first to study higher K-theory of T which translates into RV in the local situation of R being the ring of integers in a p-adic field. Before now, higher K-theory of T and hence RV has been studied only in global context of R being the ring of integers in a number field, (see [12, 7.5], [13,14]). …”

Section: Introductionmentioning

confidence: 99%

Higher AlgebraicK-Theory ofp-Adic Orders and Twisted Polynomial and Laurent Series Rings Overp-Adic Orders

Kuku

2011

Communications in Algebra

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Let F be a p-adic field (i.e., any finite extension of p ), R the ring of integers of F , any R-order in a semisimple F -algebra → an R-automorphism of T = t , the infinite cyclic group, t , the -twisted polynomial ring over and T , the -twisted Laurent series ring over . In this article, we study higher K-theory of , t , and T .More precisely, we prove in Section 1 that for all n ≥ 1, SK 2n−1 is a finite p-group if is a direct product of matrix algebra over fields, in partial answer to an open question whether SK 2n−1 p G is a finite p-group if G is any finite group. So, the answer is affirmative if p G splits.We also prove that NK n = ker K n t → K n is a p-torsion group and also that for n ≥ 2 there exists isomorphisms ⊗ K n T ⊗ G n T ⊗ K n T Finally, we prove that NK n T is p-torsion. Note that if G is a finite group and = RG such that G = G, then T is the group ring RV where V is a virtually infinite cyclic group of the form V = G T , where is an automorphism of G and the action of the infinite cyclic group T = t on G is given by g = tgt −1 for all g ∈ G.

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

Cited by 12 publications

References 6 publications

Higher Algebraic K-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

Higher Algebraic K-theory for Twisted Laurent Series Rings Over Orders and Semisimple Algebras

Splitting formulas for certain Waldhausen Nil-groups

Higher AlgebraicK-Theory ofp-Adic Orders and Twisted Polynomial and Laurent Series Rings Overp-Adic Orders

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