Group homology and extensions of groups (Q2478204)

The main result of the paper is a formula for the homology of a group \(\Pi\) with coefficients in a field of characteristic zero, as an inverse limit of a functor on the category of group extensions of \(\Pi\). The authors mention that analogs of some of the results may hold with integer coefficients. In this paper, however, they use results of \textit{D. Quillen} [``Cyclic cohomology and algebra extensions,'' K-Theory 3, No. 3, 205--246 (1989; Zbl 0696.16021)] describing the cyclic homology of an algebra over a field of characteristic zero in terms of algebra extensions. The connection with cyclic homology is via \textit{D. Burghelea}'s calculation [``The cyclic homology of group rings,'' Comment. Math. Helv. 60, 354--365 (1985; Zbl 0595.16022)] of the cyclic homology of group rings (with coefficients in a characteristic zero field), which for free groups consists of repeated copies of their group homology.