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

Let $G$ be a discrete countable infinite group. Let $T$ and $\tilde{T}$ be two rank-one $\sigma$-finite measure preserving actions of $G$ and let $\mathcal{T}$ and $\tilde{\mathcal{T}}$ be the cutting-and-stacking parameters that determine $T$ and $\tilde{T}$ respectively. We find necessary and sufficient conditions on $\mathcal{T}$ and $\tilde{\mathcal{T}}$ under which $T$ and $\tilde{T}$ 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 $\tilde{T}$ are finite measure preserving then we also find necessary and sufficient conditioins on $\mathcal{T}$ and $\tilde{\mathcal{T}}$ under which $\tilde{T}$ is a factor of $T$.