Positive harmonic functions on groups and covering spaces

Abstract

We show that if $p:M\to N$ 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}.

The previous version of the paper has been split into [P. Polymerakis, Positive harmonic functions on groups and covering spaces, Adv. Math. 379 (2021), 107552, 8 pp] and [P. Polymerakis, On the strong Liouville property of covering spaces, Potential Anal. 59 (2023), 1449-1455]. The current version is a correction of the first part, dealing with a gap pointed out by Anna Erschler