Funny rank one and the approximate transitivity for induced actions (Q1304988)

Let \(H\) be a closed normal subgroup of a locally compact separable group \(G\) acting ergodically on a Lebesgue space. It is proved, that if both the given \(H\)-section and the natural \(H\)-action on \(G/H\) have funny rank one, then the induced \(G\)-action has it too. A similar theorem is formulated for approximately transitive actions, even without the normality of \(H\). Corollaries concerning solvable group actions having discrete spectrum and fuzzy rank of a transitive action of a totally disconnected group are also provided.