Controlled algebraic \(K\)-theory of integral group ring of \(SL(3,\mathbb{Z})\)
The isomorphism-conjecture of Farrell and Jones states that the assembly map \[ \underset\text{Or}(G,{\mathcal F}{)}{\text{hocolim}} \mathbb{K}\mathbb{K}(G/G) \] in an equivalence of \(\Omega\)-spectra where \(G\) is a group, \({\mathcal F}\) is an appropriate family of subgroups, \(\text{Or}(G,{\mathcal F})\) is the orbit category with respect to \({\mathcal F}\) and \(\mathbb{K}\) is a functor from the full orbit category \(\text{Or}(G)\) to the category of \(\Omega\)-spectra. Farrell and Jones have proven this conjecture for the two-fold loops of the \(\Omega\)-spectrum given by topological pseudo-isotopy for \({\mathcal F}\) the family of virtually cyclic subgroups, provided that \(G\) is a cocompact discrete subgroup of a virtually connected Lie group. The paper deals with the computation of \(\pi_n\) of the source of the assembly map in this case for \(i=-1,0,1\), where \(G\) is a finitely generated infinite subgroup of \(SL(3,\mathbb{Z})\). It turns out to be zero for \(i=0,1\) and to be \(\mathbb{Z}^k\), where \(k\) is the number of distinct conjugacy classes of the dihedral group of order 12 in \(G\). In the first step the calculation is reduced to the family of finite subgroups and in the second step a spectral sequence argument is used.
scientific article; zbMATH DE number 929695