Groups with \(\mathbb Z\)-torsion homology. (Q2911275)

Let \(\mathcal T_H\) be the class of groups that have \(\mathbb Z\)-torsion homology after some dimension, for any coefficients. The paper under review examines the closure properties of the class \(\mathcal T_H\). It is obvious that the class \(\mathcal T_H\) contains finite groups and groups of finite cohomological dimension. Using the basic properties of the homology groups, the author proves that the class \(\mathcal T_H\) is closed under subgroups, amalgamated free products, and extensions. Using the closure properties, the author proves that almost cyclic groups (i.e. groups that admit a normal series where the successive quotients are either finite or cyclic) belong to \(\mathcal T_H\) and, in particular, \(\mathcal T_H\) contains the virtually polycyclic groups.