Absence of Non-Constant Harmonic Functions with ℓ -gradient in some Semi-Direct Products.

To obtain groups with bounded harmonic functions (which are amenable), one of the most frequent way is to look at some semi-direct products ( e.g. lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in ℓ , for $p\in [1,\infty [$ . [ABSTRACT FROM AUTHOR]