A Free Product Formula for the Sofic Dimension

It is proved that if G = G1 ∗G3 G2 is free product of probability measure preserving s-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid G3, then the sofic dimension s(G) satisfies the equality s(G) = h(G01)s(G1) + h(G 0 2)s(G2) − h(G 0 3)s(G3) where h is the normalized Haar measure on G. Let G be a group. The sofic dimension of G is an asymptotic invariant that accounts for the number of unital maps σ∶F ± → Sym(d) from the “Cayley ball” F ± of radius n in G into the symmetric group Sym(d), where F ⊂ G is a finite set, n is an integer, d is a ‘very large’ integer and the maps σ are multiplicative and free up to an error δ > 0 relative to the normalized Hamming distance on Sym(d) (see §1 below). If SA(F,n, δ, d) is the (finite) set of all such maps, and NSA ∶= ∣{σ∣F , σ ∈ SA}∣, the sofic dimension of F is: s(F ) = inf n∈N inf δ>0 lim sup d→∞ logNSA(F,n, δ, d)