Every exponential group supports a positive harmonic function

TL;DR

This paper proves that any group with exponential growth, or similar Markov chains with exponential growth, admits non-constant positive harmonic functions, extending understanding of harmonic functions in geometric group theory.

Contribution

It establishes that all exponential growth groups support non-constant positive harmonic functions, generalizing previous results to broader classes of Markov chains.

Findings

Exponential growth groups support non-constant positive harmonic functions

Results extend to certain Markov chains with exponential growth

Supports broader understanding of harmonic functions in geometric contexts

Abstract

We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, out results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive group of automorphisms for which the directed balls grow exponentially.