Extensions of residually finite groups (Q1320151)

The author investigates the problem of the residual finiteness of extensions of residually finite groups. The following theorem is proved: ``Let \(G=\text{SL}_ d(\mathbb{Z})\), and \(M =\mathbb{Z}^ d\) its natural module. (1) If \(d=2\), then every extension of \(G\) over \(M\) -- or any other residually finite \(G\)-group -- is residually finite. (2) If \(d=3\), then \(H^ 2(G,M) =\mathbb{Z}/2^ t\oplus\mathbb{Z}\), for some positive integer \(t\). Moreover, a 2-class determines a residually finite extension if and only if it is a torsion class. (3) If \(d > 3\), but \(d\neq 5\), then \(H^ 2(G,M)=0\).'' From the other results we mention only one. An extension \(E\) of \(G\) over \(M\) is said to be virtually split if there is a subgroup of finite index in \(E\) that contains and is split over \(M\). For a residually finite \(G\)-group \(M\) and an extension \(E\) of \(G\) over \(M\) the author proves that \(E\) is residually finite if and only if \(E\) is residually virtually split.