On the Spectrum and Density of States of Graphs and Groups
Abstract
We study spectra and spectral measures of the discrete Laplacian and Markov operators on the Cayley graph of a finitely generated group G (and more generally the density of states of discrete periodic operators of finite order on G-periodic graphs). Several examples of computations of spectra and spectral measures of Cayley graphs are surveyed. Some well known theorems and applications of the theory to other areas of mathematics (Kesten’s theorem, Kadison-Kaplansky conjecture, Property (T) and expander graphs) are described, following various papers, monographs and textbooks.
In the next chapter, we discuss an algebraic approach for the computation of the density of states: the use of finite support eigenfunctions, following the preprint paper of the author [1]. It is shown that eigenfunctions of λ with finite support are dense in the l2-eigenspace of λ. Moreover, if G is a virtually polycyclic finitely generated group, then there are finitely many finite support eigenfunctions of λ up to translations and linear combinations. This property can be used to approximate the density of λ. When G has subexponential growth, the density of λ is obtained from a finite resolution by finitely generated free CG-modules (if it exists) of the CG-module of finite support eigenfunctions. Such a resolution always exists when G is abelian.
In the final chapter, we discuss some examples and suggest directions for further study of the use of finite support eigenfunctions.
Subject
finitely generated groupsCayley graphs
periodic graphs
discrete Laplacian
Markov operator
spectrum
density of states
free resolution
finite support eigenfunctions
Citation
Kravaris, Cosmas (2022). On the Spectrum and Density of States of Graphs and Groups. Master's thesis, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /197393.