Measurable Splittings and the Measured Group Theoretic Structure of Wreath Products

The uploaded paper proves exactly the two statements in question under the hypothesis that Γ admits an essential measurable splitting, namely: (1) B ≀ Γ is orbit equivalent to (B × A) ≀ Γ for every amenable A, and (2) if B0 × A0 and B1 × A1 are measure equivalent (with A0, A1 amenable), then B0 ≀ Γ and B1 ≀ Γ are orbit equivalent; see Theorem 7.1 and its proof, which leverages groupoid-level bi-cofinitely equivariant constructions and an aperiodic amenable free factor guaranteed by the splitting hypothesis . The model’s solution instead reduces both claims to the paper’s “in particular” consequences (Theorem 1.3) that: (i) if B and C are measure equivalent, then B ≀ Γ and C ≀ Γ are orbit equivalent, and (ii) B ≀ Γ is orbit equivalent to (B × Z) ≀ Γ, then combines these with standard amenability/ME facts; this is logically sound and requires the same hypotheses (nontrivial base, amenable A, and Γ with an essential measurable splitting) .