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

Grigorchuk, Rostislav; Simanek, Brian

doi:10.1090/tran/8156

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators
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Trans. Amer. Math. Soc. 374 (2021), 2421-2445 Request permission

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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