arXiv subjects

Dmytro Savchuk

Publications and source records attributed to Dmytro Savchuk.

At least 19 recordsLinked to original sources

arXiv2026

Combinatorial structures connecting Latin squares and bireversible automata

This paper explores the theory of letter transducers, Mealy automata, and bireversible automata from a combinatorial perspective analogous to the theory of Latin squares. We view the sets of transitions of letter transducers as analogs of orthogonal arrays, and discuss two other combinatorial encodings of Mealy automata analogous to orthogonal pairs of Latin squares and to $(k,n)$-nets. We characterize various classes of automata (Mealy, reversible, invertible, bireversible) in terms of these combinatorial structures. In particular, we represent the inversion and dualization of transducers as parastrophisms. Further, similarly to the notion of the isotopisms of the quasigroups associated to Latin squares, we develop the notion of isotopisms of letter transducers generalizing transducer symmetry and preserving the class of bireversible automata.

cs.FL↗

arXiv2026

Contracting Self-similar Groups in Group-Based Cryptography

We propose self-similar contracting groups as a platform for cryptographic schemes based on simultaneous conjugacy search problem (SCSP). The class of these groups contains extraordinary examples like Grigorchuk group, which is known to be non-linear, thus making some of existing attacks against SCSP inapplicable. The groups in this class admit a natural normal form based on the notion of a nucleus portrait, that plays a key role in our approach. While for some groups in the class the conjugacy search problem has been studied, there are many groups for which no algorithms solving it are known. Moreover, there are some self-similar groups with undecidable conjugacy problem. We discuss benefits and drawbacks of using these groups in group-based cryptography and provide computational analysis of variants of the length-based attack on SCSP for some groups in the class, including Grigorchuk group, Basilica group, and others.

math.GR↗

arXiv2024

Liftability and Contracting Property of Multi-EGS Groups

We provide sufficient conditions for the multi-EGS groups to be liftable and thus produce new examples of groups acting transitively on regular trees of finite degree stabilizing one of the ends, whose closures are scale groups as defined by Willis. Additionally, we explicitly compute the contracting nuclei of the groups in this class. We also specialize our results to the classes of multi-edge spinal group and EGS-groups.

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

Explicit Generators for the Stabilizers of Rational Points in Thompson's Group $F$

We construct explicit finite generating sets for the stabilizers in Thompson's group $F$ of rational points of a unit interval or a Cantor set. Our technique is based on the Reidemeister-Schreier procedure in the context of Schreier graphs of such stabilizers in $F$. It is well known that the stabilizers of dyadic rational points are isomorphic to $F\times F$ and can thus be generated by 4 explicit elements. We show that the stabilizer of every non-dyadic rational point $b\in (0,1)$ is generated by 5 elements that are explicitly calculated as words in generators $x_0, x_1$ of $F$ that depend on the binary expansion of $b$. We also provide an alternative simple proof that the stabilizers of all rational points are finitely presented.

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↗

arXiv2018

The lamplighter group of rank two generated by a bireversible automaton

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group $(\mathbb Z_2^2)\wr\mathbb Z$ of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring $\mathbb Z_2[[t]]$ of formal power series over $\mathbb Z_2$.

math.GR↗

arXiv2017

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers

We show that every polynomial in $\mathbb Z[x]$ defines an endomorphism of the $d$-ary rooted tree induced by its action on the ring $\mathbb Z_d$ of $d$-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in $\mathbb Z[x]$ of the same degree. In the case of permutational polynomials acting on $\mathbb Z_d$ by bijections the induced endomorphisms are automorphisms of the tree. In the case of $\mathbb Z_2$ such polynomials were completely characterized by Rivest. As our main application we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial $f(x)\in\mathbb Z[x]$ that is necessary and sufficient for $f$ to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of $f(x)$ on $\mathbb Z_2$ with respect to the normalized Haar measure.

math.GR↗

arXiv2015

An Example of an Automatic Graph of Intermediate Growth

We give an example of a 4-regular infinite automatic graph of intermediate growth. It is constructed as a Schreier graph of a certain group generated by 3-state automaton. The question was motivated by an open problem on the existence of Cayley automatic groups of intermediate growth.

math.GR↗

arXiv2015

Ergodic decomposition of group actions on rooted trees

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and the, so called, Universal group are considered in order to demonstrate applications of the theorem.

math.GR↗

arXiv2014

Orbit automata as a new tool to attack the order problem in automaton groups

We introduce a new tool, called the orbit automaton, that describes the action of an automaton group $G$ on the subtrees corresponding to the orbits of $G$ on levels of the tree. The connection between $G$ and the groups generated by the orbit automata is used to find elements of infinite order in certain automaton groups for which other methods failed to work.

math.GR↗

arXiv2014

A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group

The class of automaton groups is a rich source of the simplest examples of infinite Burnside groups. However, there are some classes of automata that do not contain such examples. For instance, all infinite Burnside automaton groups in the literature are generated by non reversible Mealy automata and it was recently shown that 2-state invertible-reversible Mealy automata cannot generate infinite Burnside groups. Here we extend this result to connected 3-state invertible-reversible Mealy automata, using new original techniques. The results provide the first uniform method to construct elements of infinite order in each infinite group in this class.

cs.FL↗

arXiv2014

Schreier graphs of actions of Thompson's group F on the unit interval and on the Cantor set

Schreier graphs of the actions of Thompson's group $F$ on the orbits of all points of the unit interval and of the Cantor set with respect to the standard generating set $\{x_0,x_1\}$ are explicitly constructed. The closure of the space of pointed Schreier graphs of the action of $F$ on the orbits of dyadic rational numbers and corresponding Schreier dynamical system are described. In particular, we answer the question of Grigorchuk on the Cantor-Bendixson rank of the underlying space of the Schreier dynamical system in the context of $F$. As applications we prove that the pointed Schreier graphs of points from $(0,1)$ are amenable, have infinitely many ends, and are pairwise non-isomorphic. Moreover, we prove that points $x,y\in(0,1)$ have isomorphic non-pointed Schreier graphs if and only if they belong to the same orbit of $F$.

math.GR↗

arXiv2013

Self-similar groups acting essentially freely on the boundary of the binary rooted tree

We study the class of groups generated by automata that act essentially freely on the boundary of a rooted tree. In the process we establish and discuss some general tools for determining if a group belongs to this class, and explore the connections of this class to the classes of just-infinite and scale-invariant groups. Our main application is a complete classification of groups generated by 3-state automata over 2-letter alphabet that are in this class.

math.GR↗

arXiv2013

On Restricting Subsets of Bases in Relatively Free Groups

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = {a_1,..., a_r} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {a_{l+1},...,a_r}, then S is a subset of a basis for the relatively free group on {a_1,...,a_l}.

math.GR↗

arXiv2012

Submanifold Projection

One of the most useful tools for studying the geometry of the mapping class group has been the subsurface projections of Masur and Minsky. Here we propose an analogue for the study of the geometry of Out(F_n) called submanifold projection. We use the doubled handlebody M_n = #^n S^2 \times S^1 as a geometric model of F_n, and consider essential embedded 2-spheres in M_n, isotopy classes of which can be identified with free splittings of the free group. We interpret submanifold projection in the context of the sphere complex (also known as the splitting complex). We prove that submanifold projection satisfies a number of desirable properties, including a Behrstock inequality and a Bounded Geodesic Image theorem. Our proof of the latter relies on a method of canonically visualizing one sphere `with respect to' another given sphere, which we call a sphere tree. Sphere trees are related to Hatcher normal form for spheres, and coincide with an interpretation of certain slices of a Guirardel core.

math.GR↗

arXiv2011

On the geometry of a proposed curve complex analogue for $Out(F_n)$

The group $\Out$ of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which $\Out$ acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex $\ESC$, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in $\ESC$. This shows that $\ESC$: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of $\Out$. As a consequence, there is no coarsely $\Out$-equivariant quasiisometry between $\ESC$ and other proposed curve complex analogues, including the regular free splitting complex $\FSC$, the (nontrivial intersection) free factorization complex $\FFZC$, and the free factor complex $\FFC$, leaving hope that some of these complexes are hyperbolic.

math.GR↗

arXiv2008

Some graphs related to Thompson's group F

The Schreier graphs of Thompson's group F with respect to the stabilizer of 1/2 and generators x_0 and x_1, and of its unitary representation in L_2([0,1]) induced by the standard action on the interval [0,1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x_nv, where n>=0 and v is any word over the alphabet {x_0, x_1}, is constructed. It is proved that the latter graph is non-amenable.

math.GR↗