Induction and Computation of Bass Nil Groups for Finite Groups
نویسنده
چکیده
Let G be a finite group. We show that the Bass Nil-groups NKn(RG), n ∈ Z, are generated from the p-subgroups of G by induction, certain twistings maps depending on elements in the centralizers of the psubgroups, and the Verschiebung homomorphisms. As a consequence, the groups NKn(RG) are generated by induction from elementary subgroups. For NK0(ZG) we get an improved estimate of the torsion exponent.
منابع مشابه
Relating the Farrell Nil-groups to the Waldhausen Nil-groups
Every virtually cyclic group Γ that surjects onto the infinite dihedral group D∞ contains an index two subgroup Π of the form H ⋊α Z. We show that the Waldhausen Nil-group of Γ vanishes if and only if the Farrell Nil-group of Π vanishes. 1. Statement of results. The Bass Nil-groups, Farrell Nil-groups, and Waldhausen Nil-groups appear respectively as pieces in the computation of the algebraicK-...
متن کاملThe Behavior of Nil-Groups under Localization and the Relative Assembly Map
We study the behavior of the Nil-subgroups of K-groups under localization. As a consequence we obtain that the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups is rationally an isomorphism. Combined with the equivariant Chern character we obtain a complete computation of the rationalized source of the K-theoretic assembly map in terms of grou...
متن کاملAlgebraic K-theory over the infinite dihedral group: a controlled topology approach
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell–Jones conjecture can be reduced to only those virtually cyclic groups that admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil gro...
متن کاملSome Remarks on Nil Groups in Algebraic K-theory
This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Results Let R be a ring with unit. For an integer q, let KqR be the algebraic K-group of Bass and Quillen. Bass ...
متن کاملGeneralized Nil-coxeter Algebras over Discrete Complex Reflection Groups
We define and study generalized nil-Coxeter algebras associated to Coxeter groups. Motivated by a question of Coxeter (1957), we construct the first examples of such finite-dimensional algebras that are not the ‘usual’ nil-Coxeter algebras: a novel 2-parameter type A family that we call NCA(n, d). We explore several combinatorial properties of NCA(n, d), including its Coxeter word basis, length...
متن کامل