  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 in which the base group is defined via its
properly chosen Schreier graph.
