Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution

Abstract

There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators.

To appear in: Groups, Graphs, and Random Walks. T.Ceccherini-Silberstein, M.Salvatori and E.Sava-Huss Eds, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2016