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

Grigorchuk, Rostislav; Simanek, Brian

doi:10.1090/tran/8156

Cited by 6 publications

(8 citation statements)

References 34 publications

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“…In [21], Veselić studies spectral properties of percolation Hamiltonians on amenable quasi-homogeneous graphs in connection with the quantum percolation model. Finally, in [8],…”

Section: Introductionmentioning

confidence: 92%

On the density of eigenvalues on periodic graphs

Kravaris¹

2021

Preprint

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Using the Floquet-Bloch transform, we show that Z d -periodic graphs have finitely many finite support eigenfunctions up to translations and linear combinations and show that this can be used to calculate the density of eigenvalues. We study the Kagome lattice to illustrate these techniques and generalize the claims to amenable quasi-homogeneous graphs whose acting group has Noetherian group algebra (this includes all virtually polycyclic groups). Finally, we provide a formula for the von Neumann dimension of eigenvalues of Z d -periodic graphs using syzygy modules.

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“…In [21], Veselić studies spectral properties of percolation Hamiltonians on amenable quasi-homogeneous graphs in connection with the quantum percolation model. Finally, in [8],…”

Section: Introductionmentioning

confidence: 92%

On the density of eigenvalues on periodic graphs

Kravaris¹

2021

Preprint

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“…The Lamplighter map was used in [ 30 , 31 ] to describe unusual spectral properties of the Lamplighter group, and to answer the Atiyah question on

-Betti numbers [ 17 ].…”

Section: Basic Examplesmentioning

confidence: 99%

“…Surprisingly, such a simple concept did not attract much attention when the number of non-homogeneous parameters is greater or equal to two, until publication of [ 23 ]. Sporadic examples of analysis of the structure of

and in some cases of computations are presented in [ 1 , 30 , 31 , 32 , 33 , 54 , 55 , 56 , 57 ]. Important examples of pencils come from

-algebras associated with self-similar groups that were discussed in the previous section.…”

Section: Joint Spectrum and Operator Systemsmentioning

confidence: 99%

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Integrable and Chaotic Systems Associated with Fractal Groups

Grigorchuk

Samarakoon

2021

Entropy

Self Cite

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Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

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“…A union of any finite number of disjoint intervals and one or two isolated points in the spectrum can appear in the case of anisotropic Laplacians on free products of several finite cyclic groups [27]. Infinitely many gaps may appear in the spectrum of an anisotropic Laplacian on a lamplighter group [21]. The first examples of Schreier graphs whose spectrum is a Cantor set of Lebesgue measure zero or a union of such Cantor set and a countable set of isolated points accumulating on it were obtained in [1], and it is still open whether Cantor spectrum can occur on a Cayley graph.…”

Section: Introductionmentioning

confidence: 99%

On spectra and spectral measures of Schreier and Cayley graphs

Grigorchuk,

Nagnibeda,

Pérez

2020

Preprint

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We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers d ≥ 2 and m ≥ 1, we consider an uncountable family of groups of automorphisms of the rooted d-regular tree which provide examples of the following interesting phenomena. For d = 2 and any m ≥ 2, we get an uncountable family of non quasi-isometric Cayley graphs with the same Laplacian spectrum, absolutely continuous on the union of two intervals, that we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another.For each d ≥ 3, m ≥ 1, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct moreover a complete system of eigenfunctions which are strongly localized.

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

Cited by 6 publications

References 34 publications

On the density of eigenvalues on periodic graphs

On the density of eigenvalues on periodic graphs

Integrable and Chaotic Systems Associated with Fractal Groups

On spectra and spectral measures of Schreier and Cayley graphs

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