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Tatiana Nagnibeda

Publications and source records attributed to Tatiana Nagnibeda.

At least 19 recordsLinked to original sources
arXiv2025

On the ET0L subgroup membership problem in bounded automata groups

We are interested in the subgroup membership problem in groups acting on rooted $d$-regular trees and a natural class of subgroups, the stabilisers of infinite rays emanating from the root. These rays, which can also be viewed as infinite words in the alphabet with d letters, form the boundary of the tree. Stabilisers of infinite rays are not finitely generated in general, but if the ray is computable, the membership problem is well posed and solvable. The main result of the paper is that, for bounded automata groups, the membership problem in the stabiliser of any ray that is eventually periodic as an infinite word, forms an ET0L language that is constructable. The result is optimal in the sense that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be context-free. As an application, we give a recursive formula for the associated generating function, aka the Green function, on the corresponding infinite Schreier graph.

math.GR↗
arXiv2025

Non-closed subgroups of weakly branch groups

For a weakly branch group $G$ acting on a regular enough rooted tree, we provide two constructions of continuous families of distinct subgroups that are not closed in the profinite topology on $G$. On the one hand, we construct a continuous family of distinct non-closed subgroups such that each $H$ in the family is not ERF, that is, contains subgroups not closed in the profinite topology on $H$. On the other hand, under an additional assumption on $G$, we construct a continuous family of ERF subgroups which are not closed in the congruence (and in the profinite) topology on $G$.

math.GR↗
arXiv2024

Limits of Rauzy graphs of languages with subexponential complexity

To a subshift over a finite alphabet, one can naturally associate an infinite family of finite graphs, called its Rauzy graphs. We show that for a subshift of subexponential complexity the Rauzy graphs converge to the line $\mathbf{Z}$ in the sense of Benjamini-Schramm convergence if and only if its complexity function $p(n)$ is unbounded and satisfies $\lim_n\frac{p(n+1)}{p(n)} = 1$. We then apply this criterion to many examples of well-studied dynamical systems. If the subshift is moreover uniquely ergodic then we show that the limit of labelled Rauzy graphs if it exists can be identified with the unique invariant measure. In addition we consider an example of a non uniquely ergodic system recently studied by Cassaigne and Kaboré and identify a continuum of invariant measures with subsequential limits of labelled Rauzy graphs.

math.DS↗
arXiv2024

The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part

We study the planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part $\mathcal{E}_{\rm EVEN}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of $F$ isomorphic to $F$, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part $\mathcal{E}_{\rm EVEN}$ of the planar $3$-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence $\mathcal{E}_{\rm EVEN}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with $\mathcal{E}_{\rm EVEN}$: two are shown to be irreducible and one to be reducible.

math.GR↗
arXiv2024

On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$

In his work on representations of Thompson's group $F$, Vaughan Jones defined and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper we continue with the study of the $3$-colorable subgroup and prove that the quasi-regular representation of $F$ associated with the $3$-colorable subgroup is irreducible. We show moreover that the preimage of $\mathcal{F}$ under a certain injective endomorphism of $F$ is contained in three (explicit) maximal subgroups of $F$ of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of $F$, namely the parabolic subgroups that fix a point in $(0,1)$, (up to isomorphism) the Jones' oriented subgroup $\vec{F}$, and the explicit examples found by Golan.

math.GR↗
arXiv2023

Cone Types and Spectral Radius of Hyperbolic Triangle Groups and Hyperbolic Tessellations

This paper is devoted to the study of tessellations of the hyperbolic plane, especially the ones associated to hyperbolic triangle groups $Δ(l,m,n)$. We give a full description of the cone types of these graphs and show that their number depends only on the defining parameters of the group. We then use the cone types structure to provide estimates of the spectral radius for the simple random walk on these tessellations, from above and from below.

math.GR↗
arXiv2023

Spectral measures and dominant vertices in graphs of bounded degree

A graph $G = (V, E)$ of bounded degree has an adjacency operator~$A$ which acts on the Hilbert space $\ell^2(V)$. There are different kinds of measures of interest on the spectrum $Σ(A)$ of $A$. In particular, each vector $ξ\in \ell^2(V)$ defines a local spectral measure $μ_ξ$ at $ξ$ on $Σ(A)$; therefore each vertex $v \in V$ defines a vector $δ_v \in \ell^2(V)$ and the associated measure $μ_v$ on $Σ(A)$. A vertex $v$ is dominant if, for all $w \in V$, the measure $μ_w$ is absolutely continuous with respect to $μ_v$ (it then follows that, for all $ξ\in \ell^2(V)$, the measure $μ_ξ$ is absolutely continuous with respect to $μ_v$). The main object of this paper is to show that all possibilities occur: in some graphs, for example in vertex-transitive graphs, all vertices are dominant; in other graphs, only some vertices are dominant; and there are graphs without dominant vertices at all.

math.CO↗
arXiv2023

Abelian Sandpiles on Cylinders

We study here a variant of the Abelian Sandpile Model, where the playground is a cylinder of width $w$ and of circumference c. When c << w, we describe a phenomenon which has not been observed in other geometries: the probability distribution of avalanche sizes has a ladder structure, with the first step consisting of avalanches of size up to w c/2 that are essentially equiprobable, except for a small exponential tail of order about 10c. We explain this phenomenon and describe subsequent steps.

cond-mat.stat-mech↗
arXiv2022

On the oriented Thompson subgroup $\vec{F}_3$ and its relatives in higher Brown-Thompson groups

A few years ago the so-called oriented subgroup $\vec F$ of the Thompson group $F$ was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of $F$ it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, $\vec F$ provided the first example of a maximal subgroup of infinite index in $F$ different from the parabolic subgroups that fix a point in $(0,1)$. In this paper we investigate possible analogues of $\vec F$ in higher Thompson groups $F_k, k\geq 2$, with $F=F_2$, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup $\vec{F}_3$ of $F_3$, as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in $F_3$ and that the corresponding quasi-regular representation is irreducible.

math.GR↗
arXiv2022

Finitely generated subgroups of branch groups and subdirect products of just infinite groups

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups.

math.GR↗
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

Schreier graphs of spinal groups

We study Schreier dynamical systems associated with a vast family of groups that hosts many known examples of groups of intermediate growth. We are interested in the orbital graphs for the actions of these groups on $d-$regular rooted trees and on their boundaries, viewed as topological spaces or as spaces with measure. They form interesting families of finitely ramified graphs, and we study their combinatorics, their isomorphism classes and their geometric properties, such as growth and the number of ends.

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↗
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

Boundaries of $\mathbb{Z}^n$-free groups

In this paper we study random walks on a finitely generated group $G$ which has a free action on a $\mathbb{Z}^n$-tree. We show that if $G$ is non-abelian and acts minimally, freely and without inversions on a locally finite $\mathbb{Z}^n$-tree $Γ$ with the set of open ends ${\rm Ends}(Γ)$, then for every non-degenerate probability measure $μ$ on $G$ there exists a unique $μ$-stationary probability measure $ν_μ$ on ${\rm Ends}(Γ)$, and the space $({\rm Ends}(Γ), ν_μ)$ is a $μ$-boundary. Moreover, if $μ$ has finite first moment with respect to the word metric on $G$ (induced by a finite generating set), then the measure space $({\rm Ends}(Γ), ν_μ)$ is isomorphic to the Poisson--Furstenberg boundary of $(G, μ)$.

math.GR↗