English

Minimal growth harmonic functions on lamplighter groups

Probability 2016-07-05 v1 Group Theory

Abstract

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

Cite

@article{arxiv.1607.00753,
  title  = {Minimal growth harmonic functions on lamplighter groups},
  author = {Itai Benjamini and Hugo Duminil-Copin and Gady Kozma and Ariel Yadin},
  journal= {arXiv preprint arXiv:1607.00753},
  year   = {2016}
}
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