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

TL;DR

This paper constructs finitely-generated groups with Cayley graphs supporting harmonic functions of arbitrarily slow growth, demonstrating precise control over harmonic function behavior in geometric group theory.

Contribution

It introduces a method to build groups with harmonic functions of prescribed minimal growth rates using permutational wreath products and Schreier graphs.

Findings

Existence of groups with harmonic functions of growth as slow as any given sublogarithmic function.

Construction of groups where harmonic functions cannot have slower growth.

Advancement in understanding the relationship between group structure and harmonic function growth.

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