MEASURABLE IMBEDDINGS, FREE PRODUCTS, AND GRAPH PRODUCTS

The paper proves that if Λ1 SMI Γ1 and Λ2 SMI Γ2, then Λ1*Λ2 SMI Γ1*Γ2, and that the new coupling index is 1 iff both original indices are 1, and ∞ otherwise (Theorem 2.2). The construction uses a cocycle-based coupling Σ̃ = (G*Γ)×X and an explicit fundamental domain Ỹ = ({e}×X) ⊔ (W×(Y\X)), showing µ(Ỹ)=1 or ∞ depending on whether Y=X, with a careful case analysis to verify fundamental domain properties . The candidate solution also constructs a cocycle coupling Σ=Γ×X and deduces the index dichotomy via a free-product-of-equivalence-relations/treeing argument. It is mostly correct, but it contains a minor misstatement: it claims a “constant index c_i” for the subrelation S_i⊂R_i on X_i equal to µ(Y_i)/µ(X_i). In general, the subrelation index (as a class-count function) is integer-valued a.e., while µ(Y_i)/µ(X_i) need not be an integer; only the qualitative dichotomy c_i=1 vs c_i>1 is needed. Aside from this, the model’s proof is valid and achieves the same theorem by a different route.