Combinatorics of the subshift associated with Grigorchuk’s group

Abstract

Abstract. We study combinatorial properties of the subshift induced by the substitution that describes Lysenok's presentation of Grigorchuk's group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on one hand, it allowed to embed Grigorchuk's group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs. In memory of Dmitry Victorovich AnosovSubstitutional dynamical systems constitute an important class studied in symbolic dynamics. Such systems are defined by a substitution over the underlying alphabet. They appear naturally in various branches of mathematics and applications. 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. The group, defined by its action by automorphisms on the rooted binary tree, is 3-generated but not finitely presented, that is, does not admit a presentation with finitely many relations. However, Lysenok found in [7] the following recursive presentation of this group by generators and relatorsIn [4] the authors showed that the substitution appearing in Lysenok's presentation can be used to determine the spectral type of the discrete Laplacian on the Schreier graphs naturally associated with J via the action of J on the boundary of the rooted binary tree. 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.

Integrable and Chaotic Systems Associated with Fractal Groups

“…It is known that there is a subset which is both and -invariant, and the sections of which, by the lines parallel to the direction when are unions of two intervals (or an interval), while in all other directions it is a Cantor set of the Lebesgue measure zero. This follows from the results of D. Lenz, T Nagnibeda and the first author [ 25 ] and is based on the use of the substitutional dynamical system determined by substitution (see Section 12 and also [ 11 , 25 , 29 ]).…”

“…The above theorem is for the isotropic case. In the anisotrophic case by the result of D. Lenz, T. Nagnibeda and first author [ 11 , 29 ], we know only that contains a Cantor subset of the Lebesgue measure 0, which is a spectrum of a random Schrödinger operator, whose potential is ruled by the substitutional dynamical system generated by the substitution used in presentation ( 3 ).…”

“…Following [ 11 , 25 , 29 ] we consider the substitution over alphabet and system generated by it. Despite not being primitive, the system satisfies the linear repetivity property (in fact the same system can be generated by a primitive substitution ).…”

“…In our case the periods have the form . More on combinatorial properties of and associated system see [ 29 ].…”

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.…”