Positive harmonic functions on groups and covering spaces

We show that if p: M -> 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 [14]. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This study demonstrates the existence of non-constant positive harmonic functions under certain conditions.