Harmonic functions of linear growth on solvable groups

TL;DR

This paper proves that finitely generated solvable groups with finite-dimensional spaces of polynomially growing harmonic functions are virtually nilpotent, advancing understanding of group structures related to harmonic analysis.

Contribution

It confirms the conjecture for solvable groups that finite-dimensional harmonic function spaces imply virtual nilpotency, extending Kleiner's theorem.

Findings

Finite-dimensional harmonic function spaces imply virtual nilpotency in solvable groups

Supports the conjecture linking harmonic functions and group structure

Builds on Kleiner's proof for polynomial growth groups

Abstract

In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.