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Rostislav Grigorchuk

Publications and source records attributed to Rostislav Grigorchuk.

At least 19 recordsLinked to original sources
arXiv2026

On maximal subgroups of ample groups

The paper is concerned with maximal subgroups of the ample (better known as topological full) groups of homeomorphisms of totally disconnected compact metrizable topological spaces. We describe all maximal subgroups that are stabilizers of finite sets. Under certain assumptions on the ample group (including minimality), we describe all maximal subgroups that are stabilizers of closed sets or stabilizers of partitions into clopen sets. In particular, our results apply to the ample groups associated with Cantor minimal systems and to some Higman-Thompson groups.

math.GR↗
arXiv2026

Thue-Morse sequence and groups of intermediate growth

We consider the substitution subshift generated by the Thue-Morse substitution $0\to01$, $1\to10$. We prove that the topological full group of the subshift contains a subgroup of intermediate growth. Namely, one group from the family known as the Grigorchuk groups embeds into this group. To obtain our main result, we prove an embedding theorem for topological full groups, and also develop a technique to prove isomorphism of groups using the Schreier graphs.

math.GR↗
arXiv2026

Ramanujan subshifts

A finite, connected, $(d+1)$-regular graph $G$ is called Ramanujan if every its eigenvalue $λ$ satisfies either $λ=\pm (d+1)$ or $|λ|\leq 2\sqrt{d}$. The Ramanujan condition corresponds to the optimal rate of decay of correlations for the associated non-backtracking edge subshift. We consider a higher-dimensional generalization of this observation. We introduce the notion of a $d$-regular $\mathbb{Z}^δ$-subshift of finite type, and we define a Ramanujan subshift as a $d$-regular $\mathbb{Z}^δ$-subshift with an optimal rate of decay of correlations. We show that for every odd prime power $q\geq 3$ and dimension $δ<q$, there exists a $q$-regular Ramanujan $\mathbb{Z}^δ$-subshift. The construction is based on the quaternionic lattices over $\mathbb{F}_q(t)$ introduced by Rungtanapirom-Stix-Vdovina (2019). Each of our $q$-regular Ramanujan subshifts gives rise to a family of non-bipartite $(q+1)$-regular Ramanujan graphs. These graphs are very explicit and local in the strong sense: the neighbors of any vertex can be computed by an explicit Mealy automaton associated with the subshift. As a byproduct, for every odd prime power $q$, we get a single lifting rule that can be iterated to produce an infinite family of $(q+1)$-regular Ramanujan graphs.

math.DS↗
arXiv2025

Branch actions and the structure lattice

J. S. Wilson proved in 1971 an isomorphism between the structural lattice associated to a group belonging to his second class of groups with every proper quotient finite and the Boolean algebra of clopen subsets of Cantor's ternary set. In this paper we generalize this isomorphism to the class of branch groups. Moreover, we show that for every faithful branch action of a group $G$ on a spherically homogeneous rooted tree $T$ there is a canonical $G$-equivariant isomorphism between the Boolean algebra associated with the structure lattice of $G$ and the Boolean algebra of clopen subsets of the boundary of $T$.

math.GR↗
arXiv2024

Liftable self-similar groups and scale groups

We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the $(d+1)$-regular tree $\widetilde T_{d+1}$, where $d$ is the residual degree. Then we introduce the class of liftable self-similar groups acting on a $d$-regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on $\widetilde T_{d+1}$ fixing one of the ends. The closures of these extensions in $\mathrm{Aut}(\widetilde T_{d+1})$ are totally disconnected locally compact group that belong to the class of scale groups. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely or groups admitting finite $L$-presentations. In particular, we show that the finitely presented group constructed by the first author and the finitely presented HNN extension of the Basilica group embed into the group $\mathcal D(\mathbb Q_2)$ of dilations of the field $\mathbb Q_2$ of $2$-adic numbers. These actions, translated to $\widetilde T_3$, are 2-transitive on the punctured boundary of $\widetilde T_3$. Also we explore scale-invariant groups with the purpose of getting new examples of scale groups.

math.GR↗
arXiv2024

SFT covers for actions of the first Grigorchuk group

We study symbolic dynamical representations of actions of the first Grigorchuk group $G$, namely its action on the boundary of the infinite rooted binary tree, its representation in the topological full group of a minimal substitutive $\mathbb{Z}$-shift, and its representation as a minimal system of Schreier graphs. We show that the first system admits an SFT cover, and the latter two systems are conjugate to sofic subshifts on $G$, but are not of finite type.

math.DS↗
arXiv2023

Characters and IRS's on branch groups and embeddings into hyperfinite factor

Using the construction by Bencs and Tóth of invariant random subgroups on weakly branch groups acting on regular rooted trees we produce uncountably many indecomposable characters on these groups. In fact, we study three types of characters coming from the action of a weakly branch group on a regular tree, paying attention to their similarities and differences. We use obtained results to show that each countable amenable branch group has uncountably many pairwise not quasi-equivalent embeddings into Murray-von Neumann hyperfinite factor. For the canonical character associated with a self-similar group and studied by the second author as a self-similar trace we provide a number of examples when it is explicitly computed.

math.RT↗
arXiv2023

Multivariate growth and cogrowth

We investigate a multivariate growth series $Γ_L({\bf z}), {\bf z} \in \mathbb{C}^d$ associated with a regular language $L$ over an alphabet of cardinality $d.$ Our focus is on languages coming from subgroups of the free group and from subshifts of finite type. We develop a mechanism for computing the rate of growth $φ_L({\bf r})$ of $L$ in the direction ${\bf r} \in \mathbb{R}^d$. Using the concave growth condition (CG) introduced by the second author in \cite{quint2002divergence} and the results of Convex Analysis we represent $ψ_L({\bf r}) = \log\left(φ_L({\bf r})\right)$ as a support function of a convex set that is a closure of the $\textrm{Relog}$ image of the domain of absolute convergence of $Γ_L({\bf z})$. This allows us to compute $ψ_L({\bf r})$ in some important cases, like a Fibonacci language or a language of freely reduced words representing elements of a free group $F_2$. Also we show that the methods of the Large deviation theory can be used as an alternative approach. Finally, we suggest some open problems directed on the possibility of extensions of the results of the first author from \cite{grigorchuk1980symmetrical} on multivariate cogrowth.

math.GR↗
arXiv2023

Directional counting for regular languages

We explain how certain tools from convex analysis and probability theory may be used in order to obtain counting results for the number of words with prescribed frequencies of letters in regular languages.

math.CO↗
arXiv2023

On spectral properties of the Schreier graphs of the Thompson group $F$

In this article we study spectral properties of the family of Schreier graphs associated to the action of the Thompson group $F$ on the interval [0,1]. In particular, we describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to the Schreier graph $Υ$ of the orbit of 1/2. As a byproduct we calculate the asymptotics of the return probabilities of the simple random walk on $Υ$ starting at 1/2. In addition, given a Laplace type operator $L$ on a tree-like graph we study relations between the spectral measures of $L$ associated to delta functions of different vertices and the spectrum of $L$.

math.SP↗
arXiv2022

Laplace and Schrödinger operators without eigenvalues on homogeneous amenable graphs

A one-by-one exhaustion is a combinatorial/geometric condition which excludes eigenvalues from the spectra of Laplace and Schrödinger operators on graphs. Isoperimetric inequalities in graphs with a cocompact automorphism group provide an upper bound on the von Neumann dimension of the space of eigenfunctions. Any finitely generated indicable amenable group has a Cayley graph without eigenvalues. There exists a finitely generated group G with finite generating sets S and S' such that the adjacency operator of the Cayley graph of (G,S) has no eigenvalue while the adjacency operator of the Cayley graph of (G,S') has pure point spectrum.

math.SP↗
arXiv2021

Finitely generated subgroups of free groups as formal languages and their cogrowth

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

math.GR↗
arXiv2021

Integrable and Chaotic Systems Associated with Fractal Groups

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 80-s of the last century with the purpose to solve some famous problems in mathematics, including the question raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor's question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One of important developments is the relation of them to the multi-dimensional dynamics, theory of joint spectrum of pencil of operators, and spectral theory of Laplace operator on graphs. The paper gives a quick access to these topics, provide calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains discussion of the dichotomy "integrable-chaotic" in the considered model, and suggests a possible probabilistic approach to the study of discussed problems.

math.GR↗
arXiv2021

On Mealy-Moore coding and images of Markov measures

We study the images of the Markov measures under transformations generated by the Mealy automata. We find conditions under which the image measure is absolutely continuous or singular relative to the Markov measure. Also, we determine statistical properties of the image of a generic sequence.

math.DS↗
arXiv2021

Self-similar groups and holomorphic dynamics: Renormalization, integrability, and spectrum

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of discrete spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

math.GR↗
arXiv2020

On the growth of the wallpaper groups

We develop further Cannon's method of cone types for finding the growth function of a group, which can also be used to find the coordination sequences of certain infinite graphs. We then apply this method to compute the growth functions and series of the wallpaper groups (the 2 dimensional crystallographic groups). The paper has a number of illustrating colored figures and tables summarizing the results.

math.GR↗
arXiv2020

On spectra of representations and graphs. Erratum

Unfortunately the proof of the main result of [1], Theorem 1, has a flaw. Namely, Lemma 13 used in the proof of Proposition 11 is correct only under an additional assumption that the operator $A$ is normal (adjoint for the one-sided shift operator in $l^2(\mathbb N)$ provides a counterexample). Below we prove a version of Lemma 13 that does not require the normality assumption and apply it to prove Proposition 11. In addition, the same version of the lemma appears in paper [2] (as Lemma 3.1) where it is used in the proof of Theorem 1.6. We also explain here how to use the new version of Lemma 13 to correct the proof of Theorem 1.6 from [2].

math.SP↗