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) Z2 has no sublogarithmic harmonic functions, and neither has the repeated wreath product (…b(Z/22)2)…b2. These results have implications on attempts to quantify the Derriennic-Kaimanovich-Vershik theorem.

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