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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators


Authors: Rostislav Grigorchuk and Brian Simanek
Journal: Trans. Amer. Math. Soc. 374 (2021), 2421-2445
MSC (2020): Primary 47A10; Secondary 60G50, 20M35
DOI: https://doi.org/10.1090/tran/8156
Published electronically: January 26, 2021
MathSciNet review: 4223021
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Abstract:

We show that the lamplighter group $\mathcal {L}=\mathbb {Z}/2\mathbb {Z}\wr \mathbb {Z}$ has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family $a+a^{-1}+b+b^{-1} - \mu c$ of convolution operators on $\mathcal {L}$ where $\mu$ is a real parameter.

Our results show that the spectrum is a pure point spectrum for each value of $\mu$, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter $\mu$ passes the points $1$ and $-1$. Namely, if $|\mu | \leq 1$ the spectrum is an interval, while when $|\mu | > 1$ it is a union of an interval and a countable set of points accumulating to a point outside the interval.


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Additional Information

Rostislav Grigorchuk
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
MR Author ID: 193739

Brian Simanek
Affiliation: Department of Mathematics, Baylor University, Waco, Texas 76798
MR Author ID: 959574

Received by editor(s): May 22, 2019
Received by editor(s) in revised form: February 17, 2020
Published electronically: January 26, 2021
Additional Notes: The first author graciously acknowledges support from the Simons Foundation through Collaboration Grant 527814, is partially supported by the mega-grant of the Russian Federation Government (N14.W03.31.0030), and also acknowledges the Max Planck Institute in Bonn where the work on the final part of this article was completed.
Article copyright: © Copyright 2021 American Mathematical Society