On Spectra and Spectral Measures of Schreier and Cayley Graphs Academic Article uri icon

abstract

  • Abstract We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $dgeq 2$ and $m ge 1$, 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 $mgeq 2$, we get an uncountable family of non-quasi-isometric Cayley graphs with the same Laplacian spectrum, a union of two intervals, which 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 $dgeq 3, mgeq 1$, 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 that are strongly localized.

published proceedings

  • INTERNATIONAL MATHEMATICS RESEARCH NOTICES

author list (cited authors)

  • Grigorchuk, R., Nagnibeda, T., & Perez, A.

citation count

  • 3

complete list of authors

  • Grigorchuk, Rostislav||Nagnibeda, Tatiana||Perez, Aitor

publication date

  • July 2022