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 \mathcal {K} of the set of positive harmonic functions. We characterise the set of extreme points of the convex and compact set K \mathcal {K} as the set of multiplicative elements in K \mathcal {K} . Moreover, we study positive generalised eigenfunctions for a given parameter λ \lambda . We find that the topological space M λ \mathcal {M}_{\lambda } of multiplicative λ \lambda -harmonic functions is homeomorphic to a sphere for λ \lambda below a certain threshold.