On Euler classes of abelian-by-finite groups

Martin Lorenz

paper

On Euler classes of abelian-by-finite groups

arXiv:math/0112129

Abstract

Let $G$ be a finitely generated abelian-by-finite group and $k$ a field of characteristic $p \geq 0$ . The Euler class $[k_{G}]$ of $G$ over $k$ is the class of the trivial $k G$ -module in the Grothendieck group $G_{0} (k G)$ . We show that $[k_{G}]$ has finite order if and only if every $p$ -regular element of $G$ has infinite centralizer in $G$ . We also give a lower bound for the order of the Euler class in terms of suitable finite subgroups of $G$ . This lower bound is derived from a more general result on finite-dimensional representations of smash products of Hopf algebras.

12 pages, 2 figures, AMSLaTeX