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arXiv · 1409.6326

Characterisations of algebraic properties of groups in terms of harmonic functions

Matthew Tointon
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Abstract

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 V. Trofimov that an infinite vertex-transitive graph admits a non-constant harmonic function; we give a new proof of a result of T. Ceccherini-Silberstein, M. Coornaert and J. Dodziuk 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.

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Matthew Tointon. 2014-09-22. Characterisations of algebraic properties of groups in terms of harmonic functions. https://arxiv.org/abs/1409.6326

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