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Rostislav Grigorchuk

Publications and source records attributed to Rostislav Grigorchuk.

At least 37 records · Page 2Linked to original sources
arXiv2020

On spectra and spectral measures of Schreier and Cayley graphs

We are interested in various aspects of spectral rigidity of Cayley and Schreier graphs of finitely generated groups. For each pair of integers $d\geq 2$ and $m \ge 1$, we consider an uncountable family of groups of automorphisms of the rooted $d$-regular tree which provide examples of the following interesting phenomena. For $d=2$ and any $m\geq 2$, we get an uncountable family of non quasi-isometric Cayley graphs with the same Laplacian spectrum, absolutely continuous on the union of two intervals, that we compute explicitly. Some of the groups provide examples where the spectrum of the Cayley graph is connected for one generating set and has a gap for another. For each $d\geq 3, m\geq 1$, we exhibit infinite Schreier graphs of these groups with the spectrum a Cantor set of Lebesgue measure zero union a countable set of isolated points accumulating on it. The Kesten spectral measures of the Laplacian on these Schreier graphs are discrete and concentrated on the isolated points. We construct moreover a complete system of eigenfunctions which are strongly localized.

math.GR↗
arXiv2020

Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata

The ring $\mathbb Z_d$ of $d$-adic integers has a natural interpretation as the boundary of a rooted $d$-ary tree $T_d$. Endomorphisms of this tree (i.e. solenoid maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_d$ to itself and automorphisms of $T_d$ constitute the group $\mathrm{Isom}(\mathbb Z_d)$. In the case when $d=p$ is prime, Anashin showed that $f\in\mathrm{Lip}^1(\mathbb Z_p)$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a $p$-automatic sequence over a finite subset of $\mathbb Z_p\cap\mathbb Q$. We generalize this result to arbitrary integer $d\geq 2$, describe the explicit connection between the Moore automaton producing such sequence and the Mealy automaton inducing the corresponding endomorphism. Along the process we produce two algorithms allowing to convert the Mealy automaton of an endomorphism to the corresponding Moore automaton generating the sequence of the reduced van der Put coefficients of the induced map on $\mathbb Z_d$ and vice versa. We demonstrate examples of applications of these algorithms for the case when the sequence of coefficients is Thue-Morse sequence, and also for one of the generators of the standard automaton representation of the lamplighter group.

cs.FL↗
arXiv2019

On the question "Can one hear the shape of a group?" and Hulanicki type theorem for graphs

We study the question of whether it is possible to determine a finitely generated group $G$ up to some notion of equivalence from the spectrum $\mathrm{sp}(G)$ of $G$. We show that the answer is "No" in a strong sense. As the first example we present the collection of amenable 4-generated groups $G_ω$, $ω\in\{0,1,2\}^\mathbb N$, constructed by the second author in 1984. We show that among them there is a continuum of pairwise non-quasi-isometric groups with $\mathrm{sp}(G_ω)=[-\tfrac{1}{2},0]\cup[\tfrac{1}{2},1]$. Moreover, for each of these groups $G_ω$ there is a continuum of covering groups $G$ with the same spectrum. As the second example we construct a continuum of $2$-generated torsion-free step-3 solvable groups with the spectrum $[-1,1]$. In addition, in relation to the above results we prove a version of Hulanicki Theorem about inclusion of spectra for covering graphs.

math.GR↗
arXiv2019

Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphs

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

math.DS↗
arXiv2019

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

We show that the lamplighter group L has a system of generators for which the spectrum of the discrete Laplacian on the Cayley graph is a union of an interval and a countable set of isolated points accumulating to a point outside this interval. This is the first example of a group with infinitely many gaps in the spectrum of its Cayley graph. The result is obtained by a careful study of spectral properties of a one-parametric family of convolution operators on L. Our results show that the spectrum is a pure point spectrum for each value of the parameter, the eigenvalues are solutions of algebraic equations involving Chebyshev polynomials of the second kind, and the topological structure of the spectrum makes a bifurcation when the parameter passes the points 1 and -1.

math.GT↗
arXiv2019

Key-agreement based on automaton groups

We suggest several automaton groups as key-agreement platforms for Anshl-Anshel-Goldfeld metascheme, they include Grigorchuk and universal Grigorchuk groups, Hanoi 3-Towers group, Basilica group and a subgroup of the affine group with the unsolvable conjugacy problem

cs.CR↗
arXiv2018

Automatic Logarithm and Associated Measures

We introduce the notion of the Automatic Logarithm $\mathcal L_{\mathcal A, \mathcal B}$ with the purpose of studying the expanding properties of Schreier graphs of action of the group generated by two finite initial Mealy automata $\mathcal A$ and $\mathcal B$ on the levels of a regular $d$-ary rooted tree $\mathcal T$, where $\mathcal A$ is level-transitive and of bounded activity. $\mathcal L_{\mathcal A, \mathcal B}$ computes the lengths of chords in this family of graphs. Formally, $\mathcal L$ is a map $\partial \mathcal T \rightarrow \mathbb{Z}_d$ from the boundary of the tree to the integer $p$-adics whose values are determined by a Moore machine. The distribution of its outputs yields a probabilistic measure $μ$ on $\partial \mathcal T$, which in some cases can be computed by a Mealy-type machine (we then say that $μ$ is finite-state). We provide a criterion to determine whether $μ$ is finite-state. A number of examples illustrating the different cases with $\mathcal A$ being the adding machine is provided.

math.GR↗
arXiv2018

On the rigidity of rank gradient in a group of intermediate growth

We introduce and investigate the rigidity property of rank gradient in the case of the group $\mathcal G$ of intermediate growth constructed by the first author. We show that $\mathcal G$ is normally $(f,g)$-RG rigid where $f(n)=\log(n)$ and $g(n) =\log(\log(n)).$

math.DS↗
arXiv2017

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the associated rooted tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II$_1$-factor representations, and infinitely many continuous ergodic invariant random subgroups.

math.RT↗
arXiv2017

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

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.

math.GR↗
arXiv2017

Combinatorics of the subshift associated to Grigorchuk's group

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.

math.DS↗
arXiv2017

Lamplighter groups, de Bruijn graphs, spider-web graphs and their spectra

We describe the infinite family of spider-web graphs $S_{k,M,N }$, $k \geq 2$, $M \geq 1$ and $N \geq 0$, studied in physical literature as tensor products of well-known de Brujin graphs $B_{k,N}$ and cyclic graphs $C_M$ and show that these graphs are Schreier graphs of the lamplighter groups $L_k = Z/kZ \wr Z$. This allows us to compute their spectra and to identify the infinite limit of $S_{k,M,N}$, as $N, M \to\infty$, with the Cayley graph of the lamplighter group $L_k$. This is the final version of the article, taking in account comments from the referees and with an extended introduction.

math.CO↗
arXiv2017

Just-infinite C*-algebras

By analogy with the well-established notions of just-infinite groups and just-infinite (abstract) algebras, we initiate a systematic study of just-infinite C*-algebras, i.e., infinite dimensional C*-algebras for which all proper quotients are finite dimensional. We give a classification of such C*-algebras in terms of their primitive ideal space that leads to a trichotomy. We show that just-infinite, residually finite dimensional C*-algebras do exist by giving an explicit example of (the Bratteli diagram of) an AF-algebra with these properties. Further, we discuss when C*-algebras and *-algebras associated with a discrete group are just-infinite. If $G$ is the Burnside-type group of intermediate growth discovered by the first named author, which is known to be just-infinite, then its group algebra $C[G]$ and its group C*-algebra $C^*(G)$ are not just-infinite. Furthermore, we show that the algebra $B = π(C[G])$ under the Koopman representation $π$ of $G$ associated with its canonical action on a binary rooted tree is just-infinite. It remains an open problem whether the residually finite dimensional C*-algebra $C^*_π(G)$ is just-infinite.

math.OA↗
arXiv2017

On just-infinite periodic locally soluble groups

We construct an uncountable family of periodic locally soluble groups which are hereditarily just infinite. We also show that the associated full C*-algebra C*(G) is just infinite for many groups $G$ in this family.

math.GR↗
arXiv2016

On spectra of Koopman, groupoid and quasi-regular representations

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corollary of our results we obtain a continuum of pairwise disjoint pairwise equivalent irreducible representations of weakly branch groups. As an illustration we calculate spectra of regular, Koopman and groupoid representations associated to the action of the 2-group of intermediate growth constructed by the second author in 1980.

math.RT↗
arXiv2016

On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups

We study Koopman and quasi-regular representations corresponding to the action of arbitrary weakly branch group G on the boundary of a rooted tree T. One of the main results is that in the case of a quasi-invariant Bernoulli measure on the boundary of T the corresponding Koopman representation of G is irreducible (under some general conditions). We also show that quasi-regular representations of G corresponding to different orbits and Koopman representations corresponding to different Bernoulli measures on the boundary of T are pairwise disjoint. This gives two continual collections of pairwise disjoint irreducible representations of a weakly branch group. Another corollary of our results is triviality of the centralizer of G in various groups of transformations on the boundary of T.

math.RT↗
arXiv2016

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

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.

math.GR↗