Absence of non-constant harmonic functions with $\ell^p$-gradient in   some semi-direct products
  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 (\eg lamplighter groups). The aim here is to show that many of these semi-direct products do not admit harmonic functions with gradient in $\ell^p$, for $p\in [1,\infty[$. 
Antoine Gournay
math.GR math.DG math.PR