Computations of \(K\)- and \(L\)-theory of cocompact planar groups
Let \(G\) be either a cocompact planar group or an extension of the form \(1\to\mathbb{Z}^n\to G\to\pi\to 1\) where \(\pi\) is finite and acts freely on \(\mathbb{Z}^n\) outside 0. This paper contains explicit computations of the algebraic \(K\)-groups \(K_p(\mathbb{Z} G)\) for \(p\leq 1\), of the algebraic \(L\)-groups \(L_p(\mathbb{Z} G)\) for \(p\in \mathbb{Z}\) (usually with 2 inverted), and of the topological \(K\)-groups \(K_p(C^*_r)\), \(p\in \mathbb{Z}\), of the reduced \(C^*\)-algebra of \(G\). The computations apply in particular to two-dimensional crystallographic groups and to cocompact Fuchsian groups. The calculations are based on the facts that, for the groups under consideration, the finite subgroups and their normalizers are known and restricted in nature, and that the infinite virtually cyclic subgroups of cocompact groups can be listed. It is therefore possible to verify that certain assembly maps are isomorphisms, effectively confirming the Farrell-Jones and Baum-Connes isomorphism conjectures in this context.
scientific article; zbMATH DE number 1563259