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

Abstract

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 $p\in [1,\infty [$.

10 pages

References in corpus (2)

• Harmonic functions with finite $p$-energy on lamplighter graphs are constant
• A remark on the connectedness of spheres in Cayley graphs

Cited by in corpus (3)

• Harmonic functions with finite $p$-energy on lamplighter graphs are constant
• A remark on the connectedness of spheres in Cayley graphs
• Mixing, malnormal subgroups and cohomology in degree one