### Some remarks concerning harmonic functions on homogeneous graphs

*Anders Karlsson*

#### Abstract

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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