Higher \(K\)-theory of group-rings of virtually infinite cyclic groups
A group is virtually cyclic if it is either finite or virtually infinite cyclic (i.e., contains a finite index subgroup that is infinite cyclic). It was conjectured by \textit{F. T. Farrell} and \textit{L. E. Jones} [J. Am. Math. Soc. 6, 249--297 (1993; Zbl 0798.57018)], that the \(K\)-Theory of an arbitrary group \(G\) can be described in terms of virtually cyclic subgroups of \(G\). In a later paper [\(K\)-Theory 9, 13--30 (1995; Zbl 0829.19002)], \textit{F. T. Farrell} and \textit{L. E. Jones} studied the lower \(K\)-Theory of virtually infinite cyclic groups. In the present paper the authors extend this work to higher \(K\)-groups. Let \(\alpha\) be an automorphism of a ring \(A\) and \(A_\alpha[t]\) the twisted polynomial ring. Define \(NK_n(A,\alpha)\) as \(\ker(K_n (A_\alpha [t]) \to A\)) (with the map defined by \(t\mapsto 0\)). One of the most basic results is as follows: If \(R\) is the ring of integers in a number field \(F\), \(\Lambda\) an \(R\)-order in a semisimple \(F\)-algebra \(\Sigma\), and \(\alpha\) is an automorphism of \(\Lambda\), then (for \(n\geq 0\)) \(NK_n(\Lambda_\alpha[t])\) is \(s\)-torsion for some positive integer \(s\). If \(\Lambda\) is the group ring of some virtually cyclic groups then additional results on the functors \(K_n\), \(NK_n\), and \(G_n\) are obtained. For other types of virtually infinite cyclic groups, Nil groups \(NK_n\) are defined for a certain triple, with similar results.
scientific article; zbMATH DE number 1944290
scientific article