Spectra of Schreier graphs of Grigorchuk’s group and Schroedinger operators with aperiodic order
| Autor: | Tatiana Nagnibeda, Rostislav Grigorchuk, Daniel Lenz |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
Lebesgue measure Group (mathematics) General Mathematics 010102 general mathematics Spectrum (functional analysis) Boundary (topology) Mathematics::Spectral Theory 01 natural sciences Cantor set Mathematics::Group Theory Aperiodic graph 0103 physical sciences Order (group theory) 010307 mathematical physics 0101 mathematics Eigenvalues and eigenvectors Mathematics |
| Zdroj: | Mathematische Annalen. 370:1607-1637 |
| ISSN: | 1432-1807 0025-5831 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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