Characterisations of algebraic properties of groups in terms of harmonic functions (Q329576)

Summary: We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a finite-dimensional space of harmonic functions if and only if it is virtually cyclic; we present a new proof of a result of \textit{V. I. Trofimov} [Eur. J. Comb. 19, No. 4, 519--523 (1998; Zbl 0906.05028)] that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of \textit{T. Ceccherini-Silberstein} et al. [Enseign. Math. (2) 58, No. 1--2, 125--130 (2012; Zbl 1267.05182)] that the Laplacian on an infinite, connected, locally finite graph is surjective; and we show that the positive harmonic functions on a non-virtually nilpotent linear group span an infinite-dimensional space.