Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 137144
author:  Anders Karlsson 

title:  Some remarks concerning harmonic functions on homogeneous graphs 
keywords:  Discrete random walks, Dirichlet problem, radial variation, hyperbolic compactifications 
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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reference:  Anders Karlsson (2003), Some remarks concerning harmonic functions on homogeneous graphs, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 137144 
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