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) Homerton College , University of Cambridge, Hills Road, CB2 8PH, 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