Groups with minimal harmonic functions as small as you like (with an appendix by Nicolás Matte Bon)

For any order of growth f(n)=o(\log n) , we construct a finitely-generated group G and a set of generators S such that the Cayley graph of G with respect to S supports a harmonic function with growth f but does not support any harmonic function with slower growth. The construction uses permutational wreath products \mathbb{Z}/2\wr_{X}\Gamma in which the base group \Gamma is defined via its properly chosen action on X .