Harmonic functions with finite $p$-energy on lamplighter graphs are constant

Abstract

The aim of this note is to show that lamplighter graphs where the space graph is infinite and at most two-ended and the lamp graph is at most two-ended do not admit harmonic functions with gradients in ${\ell }^p$ (\ie finite $p$-energy) for any $p\in [1,\infty [$ except constants (and, equivalently, that their reduced ${\ell }^p$ cohomology is trivial in degree one). Using similar arguments, it is also shown that many direct products of graphs (including all direct products of Cayley graphs) do not admit non-constant harmonic function with gradient in ${\ell }^p$. The proof relies on a theorem of Thomassen on spanning lines in squares of graphs.

6 pages

References in corpus (3)

• Vanishing of the first reduced cohomology with values in an $L^p$-representation
• Strongly t-logarithmic t-generating sets: Geometric properties of some soluble groups
• Functions conditionally of negative type on groups acting on regular trees

Cited by in corpus (3)

• Mixing, malnormal subgroups and cohomology in degree one
• Limit theorems for random walks on Fuchsian buildings and Kac-Moody groups
• Connectedness of spheres in Cayley graphs