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Corpus ID: 119584260

Groups with minimal harmonic functions as small as you like

@article{Amir2016GroupsWM,
title={Groups with minimal harmonic functions as small as you like},
author={G. Amir and G. Kozma},
journal={arXiv: Group Theory},
year={2016}
}

For any order of growth $f(n)=o(\log n)$ we construct a finitely-generated group $G$ and a set of generators $S$ such that the Cayley graph of $G$ with respect to $S$ supports a harmonic function with growth $f$ but does not support any harmonic function with slower growth. The construction uses permutational wreath products in which the base group is defined via its properly chosen Schreier graph.