#### Volume 7, issue 4 (2007)

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Non-finiteness results for Nil-groups

### Joachim Grunewald

Algebraic & Geometric Topology 7 (2007) 1979–1986
##### Abstract

Generalizing an idea of Farrell we prove that for a ring $\Lambda$ and a ring automorphism $\alpha$ of finite order the groups ${Nil}_{0}\left(\Lambda ;\alpha \right)$ and all of its $p$–primary subgroups are either trivial or not finitely generated as an abelian group. We also prove that if $\beta$ and $\gamma$ are ring automorphisms such that $\beta \circ \gamma$ is of finite order then ${Nil}_{0}\left(\Lambda ;{\Lambda }_{\beta },{\Lambda }_{\gamma }\right)$ and all of its $p$–primary subgroups are either trivial or not finitely generated as an abelian group. These Nil-groups include the Nil-groups appearing in the decomposition of ${K}_{i}$ of virtually cyclic groups for $i\le 1$.

##### Keywords
Nil-groups, non-finiteness, twisted Laurent polynomial ring
##### Mathematical Subject Classification 2000
Primary: 18F25
Secondary: 19B28, 19D35