Rostyslav Grigorchuk - Spectrum of Schreier graphs of a group of intermediate growth, ...

Spectrum of Schreier graphs of a group of intermediate growth, Lysenok’s substitution and Schrodinger-Jacoby operators

In my talk I will consider a spectrum of a weighted Laplace operator on a Schreier graph associated with the first example G of a group of intermediate growth. It will be shown that in the degenerate case the spectrum is, as a set, a union of two intervals and is absolutely continuous with respect to the Lebesgue measure, while in the generic case it is a Cantor set of Lebesgue measure zero.
The latter result is based on the combinatorial and dynamical properties of a certain non primitive substitution τ , used for the first time by I.Lysenok to get a presentation of G by generators and
relators. This substitution has remarkable properties that will be listed.
The end of the story will be the link between a Laplace operator and a random Schrodinger-Jacoby type operator based on the use of substitution τ .
These results are based on joint results of speaker and L.Bartholdi from 2000, and recent joint results of the speaker and D.Lenz and T.Nagnibeda.