Positive harmonic functions on groups and covering spaces

Panagiotis Polymerakis

Positive harmonic functions on groups and covering spaces

Differential Geometry 2024-06-11 v3 Probability

Authors: Panagiotis Polymerakis

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}.

Keywords

harmonic functions simplicial volume function theory

Cite

@article{arxiv.1906.11723,
  title  = {Positive harmonic functions on groups and covering spaces},
  author = {Panagiotis Polymerakis},
  journal= {arXiv preprint arXiv:1906.11723},
  year   = {2024}
}

Comments

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

Positive harmonic functions on groups and covering spaces

Abstract

Keywords

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