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Spectra of Schreier graphs of Grigorchuk’s group and Schroedinger operators with aperiodic order

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Abstract

We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk’s group acting on the boundary of the infinite binary tree. We establish a connection between the action of G on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain specific graphs. The methods developed here apply to a large class of examples.

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  1. In spite of the first author’s reluctance.

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Acknowledgements

R.G. was partially supported by the NSF Grant DMS-1207669 and by ERC AG COMPASP. The authors acknowledge support of the Swiss National Science Foundation. Part of this research was carried out while D.L. and R.G. were visiting the Department of mathematics of the University of Geneva. The hospitality of the department is gratefully acknowledged. The authors also thank Yaroslav Vorobets for allowing them to use his figures 3 and 4. Finally, the authors would like to thank the anonymous referee for a careful reading of the manuscript resulting in various helpful suggestions.

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Authors and Affiliations

  1. Mathematics Department, Texas A&M University, College Station, TX, 77843-3368, USA

    Rostislav Grigorchuk

  2. Mathematisches Institut, Friedrich Schiller Universität Jena, 07743, Jena, Germany

    Daniel Lenz

  3. Section de Mathématiques, Université de Genève, 2-4, Rue du Lièvre, Case Postale, 64 1211, Geneva 4, Switzerland

    Tatiana Nagnibeda

Authors
  1. Rostislav Grigorchuk
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  2. Daniel Lenz
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  3. Tatiana Nagnibeda
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Correspondence to Daniel Lenz.

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Communicated by Thomas Schick.

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Grigorchuk, R., Lenz, D. & Nagnibeda, T. Spectra of Schreier graphs of Grigorchuk’s group and Schroedinger operators with aperiodic order. Math. Ann. 370, 1607–1637 (2018). https://doi.org/10.1007/s00208-017-1573-8

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  • DOI: https://doi.org/10.1007/s00208-017-1573-8

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