Positive Harmonic Functions on Graphs with Nilpotent Group Actions

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.