Positive harmonic functions on groups and covering spaces

TL;DR

This paper proves that non-constant positive harmonic functions exist on certain Riemannian covering spaces with exponential volume growth, confirming a conjecture by Lyons and Sullivan.

Contribution

It establishes the existence of non-constant positive harmonic functions on Riemannian coverings with exponential volume growth, resolving a conjecture.

Findings

Existence of non-constant positive harmonic functions on specific covering spaces

Confirmation of Lyons and Sullivan's conjecture

Link between volume growth and harmonic functions

Abstract

We show that if $p \colon M \to N$ is a normal Riemannian covering, with $N$ closed, and $M$ has exponential volume growth, then there are non-constant, positive harmonic functions on $M$. This was conjectured by Lyons and Sullivan in \cite{LS}.