Every word w in the free group Fr of rank r induces a probability measure (the w-measure) on every compact group G, by substitution of Haarrandom G-elements in the letters. This measure is determined by its Fourier coefficients: the w-expectations \({\mathbb E}_{w}[\chi]\) of the irreducible characters of G. For every compact group G, the wreath product with the symmetric group G ≀ Sn has some natural irreducible characters χ, and we approximate \({\mathbb E}_{w}[\chi]\) for every word w ∈ Fr, revealing new automorphism-invariant quantities of words that generalize the primitivity rank π(w). This generalizes previous works by Puder–Parzanchevski and Magee–Puder. We demonstrate applications to automorphism groups of trees, investigate properties of the new invariants, and show polynomial decay of \({\mathbb E}_{w}[\chi]\) also for wreath products with more general actions.
