On Groups of Rectangle Exchange Transformations

The paper proves that the generalized SAF map T: Rec_d → (R^{⊗(d−1)} ⊗ ∧^2_Q R)^d is surjective and that Ker(T) = D(Rec_d), hence Rec_d^{ab} ≅ (R^{⊗(d−1)} ⊗ ∧^2_Q R)^d (Theorem 1.4). It concretely defines τ_i(f) by summing the tensor-volume of translation fibers and then performs a simple coordinate change to describe the image as wedge-2 in §9.2, with restricted slab exchanges R_{i,c,a,b} mapping to simple tensors c ⊗ (a ∧ b) and spanning the target; the kernel is shown to be the derived subgroup (Proposition 9.10) . The candidate solution defines the same τ (up to the paper’s coordinate permutation σ_i) and uses the same slab two-interval exchanges to generate the image, then cites the paper’s structural result Ker(τ)=D(Rec_d). Their homomorphism check relies on the tensor-volume’s additivity/translation-invariance exactly as in the paper’s setup (Proposition 3.4) . Both arguments align on generators (restricted shuffles/rectangle transpositions) used to control the derived subgroup and abelianization (Theorems 1.2 and 1.3) .