# Groups, Geometry, and Dynamics

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**Volume 10, Issue 3, 2016, pp. 1007–1049**

**DOI: 10.4171/GGD/375**

Characterisations of algebraic properties of groups in terms of harmonic functions

Matthew C.H. Tointon^{[1]}(1) University of Cambridge, UK

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.

*Keywords: *Discrete harmonic function, discrete Laplacian, random walk, Cayley graph, linear cellular automaton

Tointon Matthew: Characterisations of algebraic properties of groups in terms of harmonic functions. *Groups Geom. Dyn.* 10 (2016), 1007-1049. doi: 10.4171/GGD/375