The Nil K-groups of finite groups

Abstract

We show that for every finite group $G$ and every integer $n$, the Nil group $\mathrm{NK}_n(\mathbb{Z}[G])$ of the integral group ring of $G$ has finite exponent, and give a bound depending on n and the order $|G|$. This bound is explicit when every prime divisor of $|G|$ is at least $3+n/2$. More generally, our finite exponent result applies to $\mathrm{NK}_n(R[G])$ whenever $R$ is a regular, torsion-free, Noetherian commutative ring such that $R/p$ is regular for every prime $p$ that divides $|G|$. In the Appendix, M. Morrow provides an alternative approach and also shows that the Nil group $\mathrm{NK}_n(X)$ of an excellent Noetherian scheme $X$, with $X[1/p]$ regular, is annihilated by a finite power of $p$, provided $X$ admits a suitable resolution of singularities. By extending his argument to certain noncommutative rings, we also show that, for $R$ as above and $\alpha$ any automorphism of $G$, the Farrell Nil groups $\mathrm{NK}_n(R[G],\alpha)$ have finite exponent. As a consequence, for every virtually cyclic group $\Gamma$, the group $\mathrm{K}_n(\mathbb{Z}[\Gamma])$ is a direct sum of a finitely generated abelian group and an infinite countable direct sum of copies of a finite abelian group.