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Subshifts with slow complexity and simple groups with the Liouville property

Geometric and Functional Analysis volume 24, pages 1637–1659 (2014)Cite this article

Abstract

We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson–Furstenberg boundary. We also get explicit upper bounds for the growth of Følner sets.

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Authors and Affiliations

  1. Département de Mathématiques et Applications, École Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France

    Nicolás Matte Bon

  2. Laboratoire de Mathématiques d’Orsay, Université Paris Sud, Faculté des Sciences d’Orsay, 91405, Orsay Cedex, France

    Nicolás Matte Bon

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  1. Nicolás Matte Bon
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Matte Bon, N. Subshifts with slow complexity and simple groups with the Liouville property. Geom. Funct. Anal. 24, 1637–1659 (2014). https://doi.org/10.1007/s00039-014-0293-4

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Keywords

  • Random Walk
  • Simple Group
  • Periodic Point
  • Topological Entropy
  • Irrational Rotation