On Euler classes of abelian-by-finite groups
Copyright policy )On Euler classes of abelian-by-finite groups
Let $G$ be a finitely generated abelian-by-finite group and $k$ a field of characteristic $p\ge 0$. The Euler class $[k_G]$ of $G$ over $k$ is the class of the trivial $kG$-module in the Grothendieck group $G_0(kG)$. 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
- Temple University United States
Homological methods in group theory, Group rings, group algebras, Smash products of general Hopf actions, Group rings of infinite groups and their modules (group-theoretic aspects), 19A31; 16S34; 16S40; 16E20; 16D90; 20J05, smash products, Grothendieck groups, Mathematics - Rings and Algebras, 20J05, 16E20, Abelian-by-finite groups, Euler classes, Rings and Algebras (math.RA), 19A31, \(K_0\) of group rings and orders, 16D90, FOS: Mathematics, Algebraic Topology (math.AT), Grothendieck groups, \(K\)-theory, etc., Mathematics - Algebraic Topology, 16S40, 16S34
Homological methods in group theory, Group rings, group algebras, Smash products of general Hopf actions, Group rings of infinite groups and their modules (group-theoretic aspects), 19A31; 16S34; 16S40; 16E20; 16D90; 20J05, smash products, Grothendieck groups, Mathematics - Rings and Algebras, 20J05, 16E20, Abelian-by-finite groups, Euler classes, Rings and Algebras (math.RA), 19A31, \(K_0\) of group rings and orders, 16D90, FOS: Mathematics, Algebraic Topology (math.AT), Grothendieck groups, \(K\)-theory, etc., Mathematics - Algebraic Topology, 16S40, 16S34
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