Topological analog of polycyclic-by-finite groups and the problem of extension of topologies (Q1204645)

A topological group \(G\) is called polycyclic-by-finite if there exists a chain \(G_ 0 \triangleleft G_ 1 \triangleleft G_ 2 \triangleleft \dots\triangleleft G_ n = G\) of subgroups where \(G_ 0\) is a compact group or a non-discrete monothetic group and \(G_ i/G_{i-1}\) is finite or cyclic and non-discrete. Theorem. Let \((G,\tau_ 0)\) be a polycyclic-by-finite topological group. Then the following conditions are equivalent: i) For every topological ring \((R,\tau_ 1)\) with 1 the topologies \(\tau_ 0\) and \(\tau_ 1\) can be extended to a ring topology of the group ring \(R[G]\). ii) The completion of the group \(G\) is a compact zero-dimensional group.