Spectra of Schreier graphs of Grigorchuk’s group and Schroedinger operators with aperiodic order

Abstract

We study spectral properties of the Laplacians on Schreier graphs arising from Grigorchuk's group acting on the boundary of the infinite binary tree. We establish a connection between the action of G on its space of Schreier graphs and a subshift associated to a non-primitive substitution and relate the Laplacians on the Schreier graphs to discrete Schroedinger operators with aperiodic order. We use this relation to prove that the spectrum of the anisotropic Laplacians is a Cantor set of Lebesgue measure zero. We also use it to show absence of eigenvalues both almost-surely and for certain specific graphs. The methods developed here apply to a large class of examples.Date: December 1, 2017.

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.…”

“…Another interesting fact observed in [8] (that also follows from [4]), is that J embeds into the topological full group of a related substitutional subshift. Various properties of this subshift were described in [4,5]. In particular, it was shown that the subshift is linearly repetitive.…”

“…While we do not need it here, we mention in passing that the sequence η can be generated by an automaton. The automaton is given in Figure 1 (see [4] for details). We define the word complexity of (Ω τ , T ) as…”

Spectra of Cayley graphs of the lamplighter group and random Schrödinger operators

“…, q, n ∈ N; p < q B 2 := {t : U k (t/2) = 0 for some k ∈ N}B 3 := 1 + 1 k : k ∈ NThe set B 3 is motivated by Proposition 5.1 and the inequalities(17).The orthogonality of the Chebyshev polynomials implies B 2 ⊆ [−1, 1]. Recall Niven's Theorem states that if x/π and sin(x) are both rational, then sin x ∈ {0, ±1/2, ±1} (see [31, Corollary 3.12]).…”