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 \(L^2\) -Betti number is multiplicative under passage to a finite-index subgroup. We give such a proof using the free uniform spanning forest (FUSF). For \(H\le\Gamma\) 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 \(I_K=1+\beta_1^{(2)}(K)\) yields \(\beta_1^{(2)}(H)=k\beta_1^{(2)}(\Gamma)\) without importing finite-index multiplicativity of von Neumann dimension.