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

Abstract

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 $X$.