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Positive harmonic functions on groups and covering spaces

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}.

Positive harmonic functions on groups and covering spaces

Abstract

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

Paper Structure

This paper contains 3 sections, 6 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Positive harmonic functions on groups
  3. Applications to Riemannian coverings

Key Result

Theorem \oldthetheorem

Let $p \colon M \to N$ be a normal Riemannian covering, with $N$ closed. If the deck transformation group of the covering has exponential growth, then there are non-constant, positive harmonic functions on $M$.

Theorems & Definitions (8)

  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Corollary \oldthetheorem