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

Mathematische Annalen (Impact Factor: 1.2). 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).


Available from: Aderemi Kuku, Jun 15, 2015
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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; DOI:10.1515/FORUM.2007.016 · 0.73 Impact Factor
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    ABSTRACT: We study Farrell Nil-groups associated to a finite order automorphism of a ring $R$. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite multiplicity. If the original finite group is a direct summand, then the countably infinite sum of the finite subgroup also appears as a direct summand. Finally, as an application, we show that if $V$ is any virtually cyclic group, then the associated Farrell or Waldhausen Nil-groups can always be expressed as a countably infinite sum of copies of a finite group.