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arXiv · 2305.01354

Positive Harmonic Functions on Graphs with Nilpotent Group Actions

Matti Richter
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Abstract

We study directed weighted graphs which are invariant under a nilpotent and cocompact group action. In particular, we consider the conic section K of the set of positive harmonic functions. We characterise the set of extreme points of the convex and compact set K as the set of multiplicative elements in K. Moreover, we study positive generalised eigenfunctions for a given parameter $\lambda$. We find that the topological space $M_\lambda$ of multiplicative $\lambda$-harmonic functions is homeomorphic to a sphere for $\lambda$ below a certain threshold.

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Matti Richter. 2023-05-02. Positive Harmonic Functions on Graphs with Nilpotent Group Actions. https://arxiv.org/abs/2305.01354

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