Classification of rank-one actions via the cutting-and-stacking parameters

Let $G$ be a discrete countable infinite group. Let $T$ and $\widetilde T$ be two rank-one $\sigma$-finite measure preserving actions of $G$ and let $\mathcal T$ and $\widetilde {\mathcal T}$ be the cutting-and-stacking parameters that determine $T$ and $\widetilde T$ respectively. We find necessary and sufficient conditions on $\mathcal T$ and $\widetilde{\mathcal T}$ under which $T$ and $\widetilde T$ are isomorphic. We also show that the isomorphism equivalence relation is a $G_\delta$-subset in the Cartesian square of the set of all admissible parameters $\mathcal T$ endowed with the natural Polish topology. If $G$ is amenable and $T$ and $\widetilde T$ are finite measure preserving then we also find necessary and sufficient conditioins on $\mathcal T$ and $\widetilde {\mathcal T}$ under which $\widetilde T$ is a factor of $T$.