Measurable splittings and the measured group theoretic structure of wreath products

Let $\mathrm{\Gamma }$ be a countable group that admits an essential measurable splitting (for instance, any group measure equivalent to a free product of nontrivial groups). We show: (1) for any two nontrivial countable groups $B$ and $C$ that are measure equivalent, the wreath product groups $B\wr \mathrm{\Gamma }$ and $C\wr \mathrm{\Gamma }$ are measure equivalent (in fact, orbit equivalent) – this is interesting even in the case when the groups $B$ and $C$ are finite; and (2) the groups $B\wr \mathrm{\Gamma }$ and $(B\times \mathbf{Z})\wr \mathrm{\Gamma }$ are measure equivalent (in fact, orbit equivalent) for every nontrivial countable group $B$. On the other hand, we show that certain wreath product actions are not even stably orbit equivalent if $\mathrm{\Gamma }$ is instead assumed to be a sofic icc group that is Bernoulli superrigid, and $B$ and $C$ have different cardinalities.