Geodesic
Some remarks concerning harmonic functions on homogeneous graphs
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003).

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We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed. 
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     author = {Karlsson, Anders},
     title = {Some remarks concerning harmonic functions on homogeneous graphs},
     journal = {Discrete mathematics & theoretical computer science},
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Karlsson, Anders. Some remarks concerning harmonic functions on homogeneous graphs. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03), DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03) (2003). doi : 10.46298/dmtcs.3348. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3348/

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