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Spectra of infinite Cayley graphs, examples with pure band spectra

Pierre de la Harpe

Abstract

It is shown that there are groups $Γ$ with finite generating sets $S$ such that the adjacency operator of the Cayley graph ${\rm Cay}(Γ,S)$ is a disjoint union of $N$ intervals, for arbitrarily large integers $N$.

Spectra of infinite Cayley graphs, examples with pure band spectra

Abstract

It is shown that there are groups with finite generating sets such that the adjacency operator of the Cayley graph is a disjoint union of intervals, for arbitrarily large integers .
Paper Structure (3 sections, 4 theorems, 3 equations)

This paper contains 3 sections, 4 theorems, 3 equations.

Table of Contents

  1. Spectrum of some operators acting on tensor products of Hilbert spaces
  2. Sums and products of graphs and Cayley graphs
  3. Examples and a proof of Theorem 1

Key Result

Theorem 1

For any $N \ge 1$, there exist Cayley graphs $G$ such that the adjacency operator $A_G$ has a pure band spectrum, consisting of $N$ disjoint intervals.

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 2
  • proof : A second proof of Proposition \ref{['PropSpecTens']}
  • Definition 3
  • Proposition 4
  • Remark 5
  • Proposition 6