Induction and computation of Bass Nil Groups for finite groups

TL;DR

This paper investigates the structure of Bass Nil-groups for finite groups, showing they can be generated from p-subgroups via induction, twisting, and Verschiebung maps, with improved torsion estimates for $NK_0(ZG)$.

Contribution

It introduces a new method to generate Bass Nil-groups from elementary subgroups using induction and specific homomorphisms, improving understanding of their structure.

Findings

Bass Nil-groups are generated from p-subgroups by induction and twisting maps.

The structure of $NK_n(RG)$ can be understood via elementary subgroups.

An improved estimate of the torsion exponent for $NK_0(ZG)$ is provided.

Abstract

Let G be a finite group. We show that the Bass Nil-groups $NK_n(RG)$, $n \in Z$, are generated from the p-subgroups of G by induction maps, certain twisting maps depending on elements in the centralizers of the p-subgroups, and the Verschiebung homomorphisms. As a consequence, the groups $NK_n(RG)$ are generated by induction from elementary subgroups. For $NK_0(ZG)$ we get an improved estimate of the torsion exponent.