Minimal growth harmonic functions on lamplighter groups

2016/07/04 by Itai Benjamini, Benjamini, Itai, Hugo Duminil-Copin +5
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #math.GR #math.PR

paper · pdf · doi:10.48550/arxiv.1607.00753

arxiv created 2016/07/04 · arxiv updated 2016/07/05

Abstract

We study the minimal possible growth of harmonic functions on lamplighters. We find that (ℤ/2)\wr ℤ has no sublinear harmonic functions, (ℤ/2)\wr ℤ² has no sublogarithmic harmonic functions, and neither has the repeated wreath product (\dotsb(ℤ/2\wrℤ²)\wrℤ²)\wr\dotsb\wrℤ². These results have implications on attempts to quantify the Derriennic-Kaimanovich-Vershik theorem.

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