On the \(K\)-theory of subgroups of virtually connected Lie groups (Q907777)

The author studies the ``assembly maps'' for finitely generated subgroups of a virtually connected (i.e. topological group which has finitely many path components) Lie group. For a given group \(G\) and a ring \(R\) the Farrell-Jones conjecture predicts the values of the algebraic \(K\)- and \(L\)-theory of the group ring \(R[G]\). One identifies this conjecture via assembly map to the much easier to handle equivariant homology groups of certain classifying spaces. Apart from this the assembly maps have geometric or analytic interpretations. Hence the Farrell-Jones Conjecture implies many very well-known conjectures; \textit{viz.} Bass, Borel, Kaplansky, and Novikov. The Farrell-Jones conjecture implies that the assembly maps \(\alpha_{\mathcal{F}in}\) for the family of finite subgroups are split injective. The rational split injectivity of the map \(\alpha_{\mathcal{F}in}\) in \(L\)-theory implies Novikov conjecture, and the integral split injectivity of \(\alpha_{\mathcal{F}in}\) is called the generalized integral Novikov conjecture. In this article, he shows that the assembly map is split injective for finitely generated subgroups of a virtually connected Lie group. This generalizes the previous result of the author over \(\mathrm{GL}_n(\mathbb{C})\). By proving the analogue result in \(L\)-theory, he could show that the discrete subgroups of virtually connected Lie groups satisfy the generalized integralNovikov conjecture.