On harmonic functions and the linear-growth case of Gromov's theorem

TL;DR

This paper establishes a characterization of groups with finite-dimensional harmonic functions, linking it to the group's algebraic structure via Gromov's theorem and polynomial growth, providing a new perspective on group analysis.

Contribution

It proves that the space of harmonic functions is finite dimensional if and only if the group has a finite-index subgroup isomorphic to the integers, using a quantitative version of Gromov's theorem.

Findings

Finite-dimensional harmonic functions characterize certain group structures.

Groups with finite-dimensional harmonic functions have a finite-index subgroup isomorphic to Z.

The proof leverages a quantitative version of Gromov's theorem on polynomial growth groups.

Abstract

We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integers. A key tool is Wilkie and van den Dries's quantitative version of the linear-growth case of Gromov's theorem on groups of polynomial growth.