Minimal growth harmonic functions on lamplighter groups

Abstract

We study the minimal possible growth of harmonic functions on lamplighters. We find that (Z/2) ≀ Z has no sublinear harmonic functions, (Z/2) ≀ Z 2 has no sublogarithmic harmonic functions, and neither has the repeated wreath productThese results have implications on attempts to quantify the Derriennic-Kaimanovich-Vershik theorem.

Shalom’s property HFD and extensions by ℤ of locally finite groups

“…In other words, all µ-harmonic functions of subexponential growth on ∆ factor through the projection ∆ → (A × B) ≀ Z. Since A × B is finite, the lamplighter group (A × B) ≀ Z does not have non-constant sublinear μ-harmonic functions [BDCKY16], it follows that ∆ doesn't have any non-constant sublinear µ-harmonic functions.…”