A Free-Uniform-Spanning-Forest Proof of Finite-Index Multiplicativity

An explicit question of Abért asks for a free-spanning-forest proof that the first L²-Betti number is multiplicative under passage to a finite-index subgroup. We give such a proof using the free uniform spanning forest (FUSF). For H ≤ Γ of index k, lift a spanning tree of the finite Schreier quotient to a forest W in a labeled Cayley multigraph G, and contract its finite components to obtain a Cayley multigraph X of H. The closed finite-cycle space of G is the graph of a bounded H-equivariant operator over that of X. Equivariant dimensions therefore give the FUSF intensity identity I_H(G) = (k−1) + I_H(X). Combining this with the FUSF formula I_K = 1 + β₁⁽²⁾(K) yields β₁⁽²⁾(H) = kβ₁⁽²⁾(Γ) without importing finite-index multiplicativity of von Neumann dimension. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.