Schreier Graphs of Grigorchuk's Group and a Subshift Associated to a Nonprimitive Substitution

Abstract

There is a recently discovered connection between the spectral theory of Schrödinger 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.

Combinatorics of the subshift associated with Grigorchuk’s group

“…In particular, certain substitutional systems provide important models for the theory of 'aperiodic order', and the spectral theory of the associated Schrödinger operators becomes a major tool in understanding the quantum mechanics of quasi-crystals [1,6]. Recently it was discovered that substitutional subshifts are also useful in the study of groups of intermediate growth [8,4,5,9].…”

“…In particular, certain substitutional systems provide important models for the theory of 'aperiodic order', and the spectral theory of the associated Schrödinger operators becomes a major tool in understanding the quantum mechanics of quasi-crystals [1,6]. Recently it was discovered that substitutional subshifts are also useful in the study of groups of intermediate growth [8,4,5,9].This note is devoted to the study of a particular substitution associated to the first group of intermediate growth constructed by the first author in 1980 in [2] and generally known as Grigorchuk's group. 1 The remarkable properties of this group described in [3] were first reported at Anosov's seminar in the Moscow State University in 1982-83.…”

Combinatorics of one-dimensional simple Toeplitz subshifts

“…The aim of this section is to give an explicit formula for the complexity function of a simple Toeplitz subshift. Our main strategy is similar to the one that was employed in [GLN17a] for the special case of the Grigorchuk subshift (see arxiv version of [GLN17b] as well): First we prove an upper bound for the complexity at certain points. Then we prove a lower bound for the growth rate of the complexity function.…”

Regularity of aperiodic minimal subshifts

“…This latter result is provided by a class of subshifts stemming from Grigorchuk's infinite 2-group G -the first known group of intermediate growth introduced by Grigorchuk [18,19] (see also [20], where a general class of groups, denoted by G ω , of intermediate growth is introduced). They have been studied, for instance, by Bon [7], Grigorchuk, Lenz and Nagnibeda [21,22], and Lenz and Sell [33]. These subshifts are determined by an infinite sequence l = (l i ) i∈N of natural numbers and we refer to them as l-Grigorchuk subshifts.…”

“…Moreover, they have also computed an explicit formula for the palindromic complexity function. Further, in the case that l is the constant one sequence, results concerning the complexity function have been obtained in [21,22].…”