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To obtain groups with bounded harmonic functions (which are not hyperbolic),one of the most frequent way is to look at some semi-direct products (\eglamplighter groups). The aim here is to show that many of these semi-directproducts do not admit harmonic functions with gradient in $\ell^p$, for $p\in[1,\infty[$.

Absence of non-constant harmonic functions with $\ell^p$-gradient in some semi-direct products