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Ievgen Bondarenko

Publications and source records attributed to Ievgen Bondarenko.

At least 19 recordsLinked to original sources
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↗
arXiv2024

The word problem and growth of groups

Let $\mathrm{WP}_G$ denote the word problem in a finitely generated group $G$. We consider the complexity of $\mathrm{WP}_G$ with respect to standard deterministic Turing machines. Let $\mathrm{DTIME}_k(t(n))$ be the complexity class of languages solved in time $O(t(n))$ by a Turing machine with $k$ tapes. We prove that $\mathrm{WP}_G\in\mathrm{DTIME}_1(n\log n)$ if and only if $G$ is virtually nilpotent. We relate the complexity of the word problem and the growth of groups by showing that $\mathrm{WP}_G\not\in \mathrm{DTIME}_1(o(n\logγ(n)))$, where $γ(n)$ is the growth function of $G$. We prove that $\mathrm{WP}_G\in\mathrm{DTIME}_k(n)$ for strongly contracting automaton groups, $\mathrm{WP}_G\in\mathrm{DTIME}_k(n\log n)$ for groups generated by bounded automata, and $\mathrm{WP}_G\in\mathrm{DTIME}_k(n(\log n)^d)$ for groups generated by polynomial automata. In particular, for the Grigorchuk group, $\mathrm{WP}_G\not\in\mathrm{DTIME}_1(n^{1.7674})$ and $\mathrm{WP}_G\in\mathrm{DTIME}_1(n^2)$.

math.GR↗
arXiv2024

The zero divisor conjecture and Mealy automata

The zero divisor conjecture is sufficient to prove for certain class of finitely presented groups where the relations are given by a pairing of generators. We associate Mealy automata to such pairings, and prove that the zero divisor conjecture holds for groups corresponding to invertible automata with three states. In particular, there cannot be zero divisors of support three corresponding to invertible pairings.

math.GR↗
arXiv2024

Quaternionic lattices and poly-context-free word problem

A finitely generated group $G$ is called poly-context-free if its word problem $\mathrm{WP}(G)$ is an intersection of finitely many context-free languages. We consider the quaternionic lattices $Γ_τ$ over the field $\mathbb{F}_{q}(t)$ constructed by Stix-Vdovina (2017), and prove that they are not poly-context-free. As a corollary, since all the groups $Γ_τ$ are quasi-isometric to $F_2\times F_2$, the class of groups with poly-context-free word problem is not closed under quasi-isometries. The result follows from the description of the language $\mathrm{WP}(Γ_τ)\cap a^*b^*c^*d^*$, which relies on the existence of anti-tori and certain power-type endomorphisms of the groups $Γ_τ$.

math.GR↗
arXiv2021

On Orbits and the Finiteness of Bounded Automaton Groups

We devise an algorithm which, given a bounded automaton A, decides whether the group generated by A is finite. The solution comes from a description of the infinite sequences having an infinite A-orbit using a deterministic finite-state acceptor. This acceptor can also be used to decide whether the bounded automaton acts level-transitively.

math.GR↗
arXiv2017

Automaton groups and complete square complexes

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three squares, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete VH complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. In particular, this gives a non-residually finite CAT(0) group isometric to $F_2\times F_2$. For the class of complete directed VH complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise on whether the main example from his PhD thesis is non-residually finite. As a by-product, we get finitely presented torsion-free simple groups which decompose into an amalgamated free product of free groups $F_7*_{F_{49}}F_7$. Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete VH complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group $F_2$ and the first automaton representation of the free product $C_3*C_3$.

math.GR↗
arXiv2016

Ends of Schreier graphs and cut-points of limit spaces of self-similar groups

Every self-similar group acts on the space $X^ω$ of infinite words over some alphabet $X$. We study the Schreier graphs $Γ_w$ for $w\in X^ω$ of the action of self-similar groups generated by bounded automata on the space $X^ω$. Using sofic subshifts we determine the number of ends for every Schreier graph $Γ_w$. Almost all Schreier graphs $Γ_w$ with respect to the uniform measure on $X^ω$ have one or two ends, and we characterize bounded automata whose Schreier graphs have two ends almost surely. The connection with (local) cut-points of limit spaces of self-similar groups is established.

math.GR↗
arXiv2014

Growth of Schreier graphs of automaton groups

Every automaton group naturally acts on the space $X^ω$ of infinite sequences over some alphabet $X$. For every $w\in X^ω$ we consider the Schreier graph $Γ_w$ of the action of the group on the orbit of $w$. We prove that for a large class of automaton groups all Schreier graphs $Γ_w$ have subexponential growth bounded above by $n^{(\log n)^m}$ with some constant $m$. In particular, this holds for all groups generated by automata with polynomial activity growth (in terms of S.Sidki), confirming a conjecture of V.Nekrashevych. We present applications to omega-periodic graphs and Hanoi graphs.

math.GR↗
arXiv2014

Finite-state self-similar actions of nilpotent groups

Let $G$ be a finitely generated torsion-free nilpotent group and $ϕ:H\rightarrow G$ be a surjective homomorphism from a subgroup $H<G$ of finite index with trivial $ϕ$-core. For every choice of coset representatives of $H$ in $G$ there is a faithful self-similar action of the group $G$ associated with $(G,ϕ)$. We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for $(G,ϕ)$. These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism $\widehatϕ$ of the Lie algebra of the Mal'cev completion of $G$.

math.GR↗
arXiv2014

The word problem in Hanoi Towers groups

We prove that elements of the Hanoi Towers groups $\mathcal{H}_m$ have depth bounded from above by a poly-logarithmic function $O(\log^{m-2} n)$, where $n$ is the length of an element. Therefore the word problem in groups $\mathcal{H}_m$ is solvable in subexponential time $\exp(O(\log^{m-2} n))$.

math.GR↗
arXiv2014

Self-similar groups and the zig-zag and replacement products of graphs

Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $Γ_n$. We construct self-similar groups, whose graphs $Γ_n$ can be represented as an iterated zig-zag product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$z$}}}Γ$ ($k\geq 1$). Also we construct self-similar groups, whose graphs $Γ_n$ can be represented as an iterated replacement product and graph powering: $Γ_{n+1}=Γ_n^k\mathop{\mbox{\textcircled{$r$}}}Γ$ ($k\geq 1$). This gives simple explicit examples of self-similar groups, whose graphs $Γ_n$ form an expanding family, and examples of automaton groups, whose graphs $Γ_n$ have linear diameters ${\rm diam}(Γ_n)=O(n)$ and bounded girth.

math.GR↗
arXiv2011

On a family of Schreier graphs of intermediate growth associated with a self-similar group

For every infinite sequence $ω=x_1,x_2,...$, with $x_i\in\{0,1\}$, we construct an infinite 4-regular graph $X_ω$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space $\{0,1\}^{\infty}$. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs $X_ω$ have intermediate growth.

math.GR↗
arXiv2011

On Lebesgue measure of integral self-affine sets

Let $A$ be an expanding integer $n\times n$ matrix and $D$ be a finite subset of $Z^n$. The self-affine set $T=T(A,D)$ is the unique compact set satisfying the equality $A(T)=\cup_{d\in D} (T+d)$. We present an effective algorithm to compute the Lebesgue measure of the self-affine set $T$, the measure of intersection $T\cap (T+u)$ for $u\in Z^n$, and the measure of intersection of self-affine sets $T(A,D_1)\cap T(A,D_2)$ for different sets $D_1,D_2\subset Z^n$.

math.MG↗
arXiv2010

Graph-directed systems and self-similar measures on limit spaces of self-similar groups

Let $G$ be a group and $ϕ:H\to G$ be a contracting homomorphism from a subgroup $H<G$ of finite index. V.Nekrashevych [25] associated with the pair $(G,ϕ)$ the limit dynamical system $(\lims,\si)$ and the limit $G$-space $\limGs$ together with the covering $\cup_{g\in G}\tile\cdot g$ by the tile $\tile$. We develop the theory of self-similar measures $μ$ on these limit spaces. It is shown that $(\lims,\si,μ)$ is conjugated to the one-sided Bernoulli shift. Using sofic subshifts we prove that the tile $\tile$ has integer measure and we give an algorithmic way to compute it. In addition we give an algorithm to find the measure of the intersection of tiles $\tile\cap (\tile\cdot g)$ for $g\in G$. We present applications to the invariant measures for the rational functions on the Riemann sphere and to the evaluation of the Lebesgue measure of integral self-affine tiles.

math.GR↗
arXiv2010

Finite generation of iterated wreath products

Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely generated if and only if the profinite abelian group $\prod_{n\geq 1} G_n/G'_n$ is topologically finitely generated. As a corollary, for a finite transitive group $G$ the minimal number of generators of the wreath power $G\wr...\wr G\wr G$ ($n$ times) is bounded if $G$ is perfect, and grows linearly if $G$ is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index, answering [2,Question 14].

math.GR↗
arXiv2008

Dynamics of piecewise linear maps and sets of nonnegative matrices

We consider functions $f(v)=\min_{A\in K}{Av}$ and $g(v)=\max_{A\in K}{Av}$, where $K$ is a finite set of nonnegative matrices and by "min" and "max" we mean coordinate-wise minimum and maximum. We transfer known results about properties of $g$ to $f$. In particular we show existence of nonnegative generalized eigenvectors for $f$, give necessary and sufficient conditions for existence of strictly positive eigenvector for $f$, study dynamics of $f$ on the positive cone. We show the existence and construct matrices $A$ and $B$, possibly not in $K$, such that $f^n(v)\sim A^nv$ and $g^n(v)\sim B^nv$ for any strictly positive vector $v$.

math.OC↗
arXiv2007

On Sushchansky p-groups

We study Sushchansky p-groups. We recall the original definition and translate it into the language of automata groups. The original actions of Sushchansky groups on p-ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain branch structures in their self-similar closures are established. We provide the connection with, so-called, G groups that shows that all Sushchansky groups have intermediate growth and allows to obtain an upper bound on their period growth functions.

math.GR↗